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XI PHYSICS

[SCALARS & VECTORS] CHAPTER NO. 2 To understand different concepts of Physics, a mathematical approach is used called as VECTORS. Vectors have solved our problems related to free body diagrams, plotting trajectories and so on. In further chapters (3, 4 & 5) we will be applying these concepts of vectors.

M. AFFAN KHAN LECTURER – PHYSICS, AKHSS, K affan_414@live.com https://promotephysics.wordpress.com

SCALARS AND VECTORS Chap#2

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SCALAR QUANITITES:

‘’Those physics quantities which are described

completely with magnitude and unit are called scalar

quantities.’’

Examples: ____________________, ____________________, ____________________, ____________________, ____________________ … etc.

VECTOR QUANTITIES:

‘’Those physics quantities which are described

completely with magnitude and unit are called scalar

quantities.’’

Examples: ____________________, ____________________, ____________________, ____________________, ____________________ … etc.

REPRESENTATION OF A VECTOR

GRAPHICAL METHOD:

Vectors are generally represented by an arrow head, whose length defines the magnitude and the head terminal defines the direction of that particular vector.

Examples:

ANALYTICAL OR COMPONENT METHOD:

In this type vector is represented in terms of component form. Usually vectors are represented in three rectangular components i.e. along x-axis (i), along y-axis (j), along z-axis (k).

Examples:

A = Axi + Ayj + Azk

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Naming of Vectors:

Vectors are usually named by small or capital letters with an arrow head. (E.g. A , F etc) or we can also write them in bold letters. (E.g. 𝐀, 𝐅)

Note: From now I will write the vector symbols in bold letters.

SPECIAL KIND OF VECTORS

:

The vector which is used to indicate or locate the position of some point from a reference is called as position vector. It is normally represented by 𝐫.

:

A vector which can be moved anywhere in space parallel to itself is called as .

:

A vector whose magnitude is zero is called as .

:

A vector which is actually the sum of two or more vectors is called as .

:

Those vectors which are use to specify only the direction of vector but their magnitude is only one are

called as . These vectors are usually written in small letters with a cap on

their head.

E.g. 𝐀 = |A|a (Here a is the which shows the direction of vector 𝐀)

Question: Discuss the significance of unit vectors in terms of i, j & k and write them in the mathematical form.

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Problem#1: Find the direction of the vector A = 3i + 6j − 2k whose magnitude is 7 units.

Problem#2: Find out the real vector whose magnitude |B| =

√38 units and the directional vector is given as

b=3i-2j+5k

√38.

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RESOLUTION OF VECTORS

‘’Those physics quantities which are described

completely with magnitude and unit are called scalar

quantities.’’ _____

It is the reverse of addition of vectors. Vector can be resolved in many components but usually they are resolved in two or three components which are perpendicular to each other. They are called as Rectangular Components of Vector.

Mathematical Form:

Consider a vector 𝐀 in a xy-plane making an angle θ with the x-axis (horizontal). Now draw two projections from the head of the vector on each axis as shown in figure. The component which is parallel to x-axis is called as horizontal component (𝐀𝐱) and the component parallel to y-axis is called as vertical component (𝐀𝐲).

We can now draw a right angled triangle from these three vectors. To find the values of these components we can use trigonometric ratios.

For Horizontal Component

We know that

cos θ = − − −

cos θ = − − −

Ax = Acos θ

For Vertical Component

We also know that

sin θ = − − −

sin θ = − − −

Ay = A sin θ

VECTOR’S MAGNITUDE & DIRECTION USING COMPONENTS

If we know the magnitude of the components of a vector we can easily find the magnitude of vector as well as direction.

Magnitude:

By applying Pythagoras theorem on the above figure we may write,

(___)2 = (___)2 + (___)2

___2 = Ax2 + Ay

2

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___ = √Ax2 + Ay

2

Similarly, in case of three dimensions

A = √_____2 + _____2 + _____2

Direction:

Direction is actually the angle made by the vector with axis. (In this case the angle is made with x-axis)

_________ =Perp

Base

_________ = __________

θ = _________ (Ay

Ax)

Problem#3: Find the unit vector parallel to the vector A =3i + 6j − 2k.

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Problem#4: Find the rectangular components of a vector A, 15 units long when it form an angle with respect to +ve x-axis of

a) 500 b) 1300 c) 3100

Problem#5: If A = i + j + k and B = 2i − j + 3k, find a unit vector parallel to A − 2B.

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Problem#6: An aircraft takes off at an angle 600 to the horizontal. If the component of the velocity along the horizontal is 200 m/s, what is its actual velocity? Find also the vertical component of its velocity.

Problem#7: If one of the rectangular component of 50 N is 25 N find the value of other.

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Problem#8: Two vectors have magnitudes 4 & 5 units. The angle between them is 300 taking first vector along x axis. Calculate the magnitude and the direction of the resultant.

Problem#9: A car weighing 10,000 N on a hill which makes an angle of 200 with the horizontal. Find the components of car’s weight parallel and perpendicular to the road.

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ADDITION OF VECTORS

There are some mathematical operations which can be applied on vectors, in which one of the operations is addition. Vector addition can be done by the following methods.

Head to tail Rule:

It is a graphical method; by this method we can add two or more vectors graphically. We can add vectors by joining the head of 1st vector with the tail of next vector and so on. And the resultant would be drawn from the tail of first vector to the head of last vector as shown in figure.

Activity: First label the given vectors, then arrange the given vectors in head – to – tail manner and find out the resultant.

Triangle Method:

Two vectors may be added by triangle method. Two vectors are connected in head to tail manner as two sides of triangle and the resultant is drawn as the third side.

Parallelogram Method:

In this method two vectors are represented as the two adjacent sides of a parallelogram and the diagonal will be considered as the resultant of these two vectors. Again this method is also applicable with two vectors’ case.

Rectangular Component Method:

By this method we can add two or more vectors by resolving them into rectangular components. The resultant would be the sum of the corresponding components of each vector.

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PROPERTIES OF VECTOR ADDITION

Commutative Property:

If same result is obtained while changing the orders of vectors to be added in addition or in multiplication, the property is called as commutative property. Commutative property of Vector Addition can be proved by Parallelogram Method.

𝐀 + 𝐁 = 𝐁 + 𝐀

Proof:

Consider two vectors A & B to be added. Draw A & B as the two adjacent sides of parallelogram as shown in figure. The resultant R would be the sum of A & B.

You can see from both sides it is clear that the resultant is always R either we add A to B or B to A.

Associative Property:

While adding, making group of two vectors out of three in such a manner that group vectors should be added first then the remaining vector should be added. If the result is same for different group, then we may say that vector addition shows associative property.

(𝐀 + 𝐁) + 𝐂 = 𝐀 + (𝐁 + 𝐂)

Proof:

Consider three vectors A, B & C to be added.

If the vector addition has associative property, then it must satisfy the following relations

(𝐀 + 𝐁) + 𝐂 = 𝐑

𝐀 + (𝐁 + 𝐂) = 𝐑

i) Adding A & B first then joining the resultant of A & B with the vector C, we will get the resultant R.

ii) Now adding B & C first then joining the resultant of B & C with the vector A, we will again get the resultant R.

It is clear from the both the graphical representation that the Vector Addition has Associative Property.

(i)

(ii)

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ADDITION OF VECTORS BY RECTANGULAR COMPONENT METHOD

Consider two vectors 𝐀𝟏 and 𝐀𝟐 aligned at angles θ1 and θ2 respectively to the horizontal (x-axis). The two vectors can be added to give a resultant vector 𝐑 which itself has an angle θ with the horizontal.

First resolve each vector into its components and then it will be clear from the figure that

𝐑𝐱 = _______________

𝐑𝐲 = _______________

But from trigonometric ratios we can recall that

𝐀𝟏𝐱 = 𝐀𝟏 𝐜𝐨𝐬 𝛉𝟏, 𝐀𝟐𝐱 = _______________

𝐀𝟏𝐲 = ______________, 𝐀𝟐𝐲 = 𝐀𝟐 𝐬𝐢𝐧 𝛉𝟐

After substituting these values in above equation we get

The sum of all x-components would be

𝐑𝐱 = 𝐀𝟏𝐱 + 𝐀𝟐𝐱 = ____________________________

And the sum of all y components can be written as

𝐑𝐲 = 𝐀𝟏𝐲 + 𝐀𝟐𝐲 = ____________________________

Magnitude of Resultant:

Magnitude of result can be written with the help of Pythagoras theorem,

(Hyp)2 = (Base)2 + (Perp)2

_____2 = _____2 + _____2

_____2 = (_____________ + _______________)2 + (_______________ + _______________)2

_____

= √(_______________ + _______________)2 + (_______________ + _______________)2

For ‘n’ vectors:

R = (A1cosθ1 + A2cosθ2 + ⋯+ Ancosθn)2

+ (A1sinθ1 + A2sinθ2 + ⋯+ Ansinθn)2

Direction of the Resultant Vector:

θ = tan−1 (Ry

Rx

)

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MULTIPLICATION OF VECTOR BY A SCALAR QUANTITY

If we multiply a vector by a scalar quantity, we always get vector as a result. For this,

1) If a vector is multiplied by a positive scalar number than its magnitude increases. However, the direction remains same.

m × 𝐀 = m𝐀

2) If a vector is multiplied by a negative scalar number than its magnitude increases but direction of resultant vector is reversed.

−m × 𝐀 = −m𝐀

3) If a vector is multiplied by zero, then the resultant is a null vector.

0 × 𝐀 = 𝐎

MULTIPLICATION OF VECTOR BY A VECTOR QUANTITY

Scalar Product or Dot Product:

“When a vector is multiplied by a vector and the resultant is a scalar quantity then this type of product is called as Scalar

Product or Dot Product. It is represented by a dot (.) between two vectors.”

Examples: __________, __________, __________ … etc.

Consider two vectors A & B having angle θ between them

𝐀. 𝐁 = Scalar Quantity

𝐀. 𝐁 = (Magnitude of 𝐀)(Projection of 𝐁 onto 𝐀)

𝐀. 𝐁 = ABA

From diagram we may see that BA = B cos θ

We get

𝐀. 𝐁 = AB cos θ

In terms of rectangular components

𝐀. 𝐁 = AxBx + AyBy + AzBz

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Properties of Dot Product:

Dot product is commutative __________ = __________

Dot product is distributive 𝐀. (𝐁 + 𝐂) = ________________________

Dot product is maximum when θ = 00, since cos00 = 1 and 𝐀. 𝐁 = AB

𝐢. 𝐢 = 𝐣. 𝐣 = 𝐤. 𝐤 = 1

Dot product is zero when θ = 900, since cos900 = 0 and then 𝐀. 𝐁 = 0

𝐢. 𝐣 = 𝐣. 𝐤 = 𝐤. 𝐢 = 0

COMMUTATIVE PROPERTY IN DOT PRODUCT OR

SHOW THAT 𝐀.𝐁 = 𝐁. 𝐀

Consider two vectors A & B having angle θ between them. The dot product between these two vectors can be written as

𝐀. 𝐁 = ABA

But BA = Bcosθ, so that

𝐀. 𝐁 = ABcosθ --------------- (1)

Similarly, if we change the order of multiplication i.e. if we write

𝐁.𝐀 = BAB

But from diagram it is clear that AB = Acosθ, so that

𝐁.𝐀 = BAcosθ

𝐁.𝐀 = ABcosθ --------------- (2)

Comparing (1) & (2)

𝐀. 𝐁 = 𝐁. 𝐀

Thus it is clear from the result that Dot product shows commutative property.

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Problem#10: Find the work done in moving an object along a vector r = 3i + 2j − 5k if the applied force is F = 2i − j − k Problem#11: Evaluate the scalar product of the following:

a) i. i b) i. k c) k. (i + j) d) (2i − j + 3k). (3i + 2j − k)

(i − 2k). (j + 3k)

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Problem#12: Find the angle between A and B where A = 6i +6j − 3k and B = 2i + 3j − 6k.

Problem#13: Find the projection of the vector A = i − 2j + k onto the direction of vector B = 4i − 4j + 7k.

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Problem#14: Find the value of p for which the following vectors are perpendicular to each other. A=i-pj+3k and B=3i+3j–4k.

DISTRIBUTIVE PROPERTY OF DOT PRODUCT

𝐀. (𝐁 + 𝐂) = 𝐀. 𝐁 + 𝐀. 𝐂

Consider three vectors A, B & C in xy-plane such that vector A lies on horizontal x-axis whereas B & C are making some angle with the horizontal.

To prove our result let us first add vectors B & C, and let the resultant be R

𝐑 = 𝐁 + 𝐂

Now taking dot product between A & R, we may write

𝐀. 𝐑 = ARA

From figure it is clear that RA = _______________

Put in above equation

𝐀. 𝐑 = A(BA + CA)

𝐀. 𝐑 = ABA + ACA

From the definition of dot product ABA = 𝐀.𝐁, ACA = 𝐀. 𝐂

Therefore, 𝐀. 𝐑 = 𝐀. 𝐁 + 𝐀. 𝐂

𝐀. (𝐁 + 𝐂) = 𝐀. 𝐁 + 𝐀. 𝐂

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Vector product or Cross Product:

“When a vector is multiplied by another vector and the resultant is a vector then this type of product is called as Vector product

or Cross Product. It is represented by a cross sign (x).”

𝐀 × 𝐁 = Vector Quantity

Examples: ____________________, _____________________,

____________________ … etc.

Mathematically

Consider two vectors A & B having an angle θ between them then the vector product can be written mathematically as,

𝐀 × 𝐁 = ABsinθu

Where u is the unit vector representing the direction of the resultant vector which always perpendicular to the plane of A & B.

In magnitude form,

|𝐀 × 𝐁| = _______________

If A & B are written in component form, like

𝐀 = Ax𝐢 + Ay𝐣 + Az𝐤

𝐁 = Bx𝐢 + By𝐣 + Bz𝐤

Then cross product can also be written as,

𝐀 × 𝐁 = |

i j kAx Ay Az

Bx By Bz

|

Direction:

The direction of cross product can be predicted by right hand rule or screw rule.

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𝐢

𝐣

𝐤

Properties of Vector Product:

Cross product doesn’t commute 𝐀 × 𝐁 ≠ 𝐁 × 𝐀 or 𝐀 × 𝐁 = −𝐁 × 𝐀

Cross product shows distributive property 𝐀 × (𝐁 + 𝐂) = _________________________

Cross product would be zero if the vectors are parallel (i.e. θ = 00), then sin00 = 0 and the cross product would be 𝐀 × 𝐁 = 0.

𝐢 × 𝐢 = 𝐣 × 𝐣 = 𝐤 × 𝐤 = 0

Cross product would be maximum if the angle between vectors is 900, then sin900 = 1, and the cross product would be |𝐀 × 𝐁| = AB

𝐢 × 𝐣 = 𝐤, 𝐣 × 𝐢 = −𝐤

𝐣 × 𝐤 = ___, 𝐤 × 𝐣 = ___

𝐢 × 𝐤 = ___, 𝐤 × 𝐢 = ___

Problem#15: Determine a unit vector perpendicular to the plane containing A and B if A = 2i − 3j − k, B = i + 4j −2k.

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CROSS PRODUCT DOESN’T COMMUTE OR

PROVE THAT 𝐀 × 𝐁 ≠ 𝐁 × 𝐀 OR

PROVE THAT 𝐀 × 𝐁 = −𝐁 × 𝐀

Consider two vectors A & B having angle θ between them. If we take cross product of these two vectors, then we may write

𝐀 × 𝐁 = ABsinθu --------------- (1)

If we change the order of multiplication then,

𝐁 × 𝐀 = ABsinθ(−u)

Multiply -1 on both sides

−(𝐁 × 𝐀) = ABsinθu --------------- (2)

Comparing (1) & (2) we get

(𝐀 × 𝐁) = −(𝐁 × 𝐀)

SHOW THAT THE MAGNITUDE |𝐀 × 𝐁| IS THE AREA OF PARALLELOGRAM

Consider two vectors A & B represented as two adjacent sides of parallelogram as shown in figure,

As we know that,

Area of parllelogram = Base × Height

From figure base is the magnitude of vector A and for height (h) draw a projection perpendicularly from the head of vector B to A.

Area of parallelogram = Ah But h = Bsinθ

Area of parallelogram = ABsinθ But since, ABsinθ = |𝐀 × 𝐁|

Therefore, Area of parallelogram = |𝐀 × 𝐁|

Hence it is proved that the area of parallelogram is actually the magnitude of cross product between two vectors.

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Problem#16: Find the area of parallelogram if its two sides are formed by the vectors A = 2i −3j − k and B = i + 4j − 2k.

Problem#17: Two sides of a triangle are formed by the vector A = 3i + 6j − 2k and vector B =4i − j + 3k. Determine the area of the triangle.

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Problem#18: Two vectors of magnitude 20 cm and 10 cm make an angle of 200 and 500 respectively with x-axis in the xy-plane; find the magnitude and direction of their cross product.

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THEORETICAL QUESTIONS (PAST PAPERS)

1. Can the magnitude of resultant of two vectors of the same magnitude be equal to the magnitude of either of the same vectors? Explain mathematically. (2011)

2. Two forces F1 and F2 are acting on a point making angles θ1 and θ2 with positive x-axis respectively. Derive the expression for the magnitude of the resultant force and its direction with respect to the positive x-axis. (2011), (2000), (1993)

3. Show that the cross product of a vector is not commutative –B × A = A × B and prove that the magnitude of cross product of two vectors gives the area of parallelogram. (2010)

4. Define the product of two vectors. Show that: A.(B+C) =A.B +A.C. (2009) (2007)

5. Can the resultant of two vector of the same magnitude be equal to the magnitude of either of the vector? Give mathematical reason of your answer. (2009)

6. Define the following: (2005) a) Unit vector b) Position vector c) Free vector

7. How many methods of addition of vectors are given in your book? Write their names. Describe the addition of two vectors A1 and A2, making angles Ө1 and Ө2, with +ve x-axis respectively by rectangular components method. (2006), (2004)

8. Define vector product of two vectors and show that (AxB) = - (BxA) (2003)

9. Show that A.(B+C) =A.B+A.C (pre-med 2002)

10. Explain the addition of two vectors by rectangular components method. Calculate the resultant force. (Premed 2002)

11. If A and B represent the adjacent sides of parallelogram. Show that |AxB| represents area of the parallelogram. (2002)

12. What are dot products and cross product? Give their properties and examples. (1999)

13. Prove that A.B =B.A, AxB= - (BxA), and A.(B+C) = A.B + A.C (1999)

14. Describe the addition of vectors by rectangular components method (1998)

15. Prove that |AxB|2 + (A.B) =A2B2 (1998)

16. Using the law of vector product prove the ‘law of Sine’s for a plane triangle of sides a, b, c. (1996)

17. Define addition of vectors by rectangular components method (1996)

18. Define vector product of two vectors. If vectors A and B are inclined at an angle of 0 degree with respect to each other, show that A x B= -B x A (1995)

19. Explain commutative and distributive law for dot product. (1994)

20. If A= A1i + A2 j + A3k, B = B1i + B2j + B3k then prove that A.B=A1B1 + A2B2 +A3B3. (1994)

21. What are vector and scalar quantities? (1993)

22. Define the Scalar Product of two vectors. What are the properties of Scalar Product? Give at least one example of scalar product. (1992)

23. What do you understand by dot product and cross product of two vectors? Explain. Give at least one example of each product. (1991)

24. Will the value of a vector quantity change if its reference axes are changed? Explain. (1991)

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NUMERICALS FROM PAST PAPERS

Magnitude of a vector, unit vector parallel, vector addition: 1. The following forces act on a particle P? F1 = 2i + 3j – 5k, F2 = -5i + j + 3k, F3 = i – 2j + 4k, F4 = 4i – 3j

– 2k Measured in Newton’s find, a) The resultant of the forces b) The magnitude of the resultant force

2. If A = 3i – j – 4k, B = -2i + 4j – 3k and C = i + 2j – k, find, a) 2A - B + 3C b) |A + B + C| c) |3A – 2B + 4C| d) a unit vector parallel to 3A – 2B + 4C

3. The position vectors of points P and Q are given by, r1 = 2i + 3j – k, r2 = 4i–3j+2k. Determine PQ in terms of rectangular unit vector, i, j and k and find its magnitude.

4. Find the rectangular components of a vector A, 15 units long when it forms an angle with respect to +ve x-axis of a) 500 b) 1300 c) 2300 d) 3100

5. Find the unit vector parallel to the vector. A = 3i + 6j – 2k. 6. If A = i + j + k and B = 2i – j + 3k, find a unit vector parallel to A – 2B. 7. Given r1 = 2i – 2j+k, r2 = 3i - 4j – 3k, r3 = 4i + 2j + 2k find the magnitude of the following vectors.

a) r3 b) r1 + r2 + r3

c) 2r1 – 3r2 – 5r3 RECTANGULAR COMPONENT METHOD: 1. An aero plane takes off at an angle 60° to the horizontal. If the component of the velocity along the

horizontal is 200 m/s, what is its actual velocity? Find also the vertical component of its velocity. (1990)

2. Two vectors have magnitudes 4 and 5 units. The angle between them is 30o taking the first vector along x axis. Calculate the magnitude and the direction of the resultant. (1997)

3. If one of the rectangular components of 50 N is 25 N find the value of other. (2010) 4. Find the work done by a force of 30,000 N in moving an object through a distance of 45 m and also

find the rate at which the force is working at a time when the velocity is 2 m/s for the following, a) The force is in the direction of motion b) The force makes an angle of 400 to the direction of motion.

5. A car weighing 10,000 N on a hill which makes an angle of 200 with the horizontal. Find the components of car’s weight parallel and perpendicular to the road.

CROSS PRODUCT: 1. Determine the unit vector perpendicular to the plane containing A and B, if A = 2i – 3j – k, B = i + 4j –

2k. (2014) 2. A = 2i – j + 3k and B = i – j + 4k are two vectors. Find a vector perpendicular to both A and B.

(2014S) 3. Two vectors of magnitude 20 cm and 10 cm make angles of 20° and 50° respectively with x-axis in the

xy plane; find the magnitude and direction of their cross product. (1991) 4. Determine a unit vector perpendicular to the plane containing A and B if A = 2i - 3j - k, B = i + 4j – 2k.

(1999) 5. Determine a unit vector perpendicular to the plane containing A and B if A = 2i – 3 j - k and B = i + 4j

- 2k. (2006) 6. If P = 2i - 2j + 3k and Q = 3i +3j + 3k find a unit vector perpendicular to the plane containing both P

and Q. if P and Q from the sides of a parallelogram, find the area of the parallelogram. (2002M) 7. Find the area of a parallelogram if its two sides are formed by the vectors, A = 2i - 3j - k and B = i+4j-

2k. (2003E) 8. If A = 2i – 3j – k, B = i + 4j – 2k, find;

a) A x B b) B x A c) (A + B) x (A - B)

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9. Determine the unit vector perpendicular to the plane of A = 2i – 6j – 3k and B = 4i + 3j – k. (2011) 10. If r1 and r2 are the position vectors (both lie in xy plane) making angle θ1 and θ2 with the positive x-

axis measured counter clockwise, find their vector product when, a) |r1| = 4cm θ1 = 300, |r2| = 3 cm θ2 = 900 b) |r1| = 6cm θ1 = 2200, |r2| = 3 cm θ2 = 400

c) |r1| = 10cm θ1 = 200. |r2| = 9 cm θ2 = 1100

11. r1 and r2 are two position vectors making angle θ1 and θ2 with positive x-axis respectively. Find their vector product when r1 = 4cm, and r2 = 3cm, θ1 = 300 and θ2 = 900.

12. Two sides of a triangle are formed by the vector A = 3i + 6j – 2k and vector B = 4i – j + 3k. Determine the area of the triangle.

DOT PRODUCT: 1. Find the angle between A and B where A = 6i + 6j -3k and B = 2i + 3j - 6k. (1992) 2. Find the angle between A = 2i + 2j – k and B = 6i – 3j + 2k. 3. Find the projection of the vector, A = i – 2j + k onto the direction of vector B = 4i – 4j + 7k. 4. Find the work done in moving an object along a vector r = 3i + 2j – 5k if the applied force is F = 2i – j

– k. 5. Evaluate the scalar product of the following:

a) i.i b) i.k c) k.(i + j) d) (2i – j + 3k).(3i + 2j - k) e) (i – 2k).(j + 3k) Where i, j, and k represents unit vectors along x, y and z axes of three dimensional rectangular coordinate system.

6. Find, a) the projection of A = 2i – 3j + 6k onto the direction of vector, B = i + 2j + 2k, b) Determine the angle between the vectors A and B.

7. Find the work done in moving an object along a straight line from (3, 2, -1) to (2, -1, 4) in a force field which is given by F = 4i – 3j + 2k and also find the angle between force and displacement.

8. An object moves along a straight line from (3, 2, -6) to (14, 13, 9) when a uniform force F = 4i + j + 3k acts on it. Find the work done and the angle between force and displacement. (2001)

9. Find the value of p for which the following vectors are perpendicular to each other. A=i-pj+3k and B=3i+3j–4k. (2009) (1996)

10. Find the angle between A = 2i + 2j – k and B = 6i -3j + 2k. (2005) 11. If A= 3i+ j - 2k; B = -i +3j + 4k find |A + B| an angle between A and B. (2000) 12. A = 3i + j – 2k and B = -i + 3j + 4k. Find the projection of A onto B. (2013)

OTHER PROBLEMS 1. If two vectors A and B are such that |A| = 3, |B| =2 and |A – B|= 4, evaluate

a) A.B b) |A + B|. (2012) (1995)

2. The angle between the vector A and B is 600. Given that |A| = |B| = 1, calculate, a) |B–A| b) |B+A|

3. Two vectors A and B are such that |A| = 3 and |B| = 4, and A.B = -5, find a) the angle between A and B b) the length |A + B| and |A – B| c) the angle between (A + B) and (A - B)

4. Two vectors A and B are such that |A| = 4, |B| = 6 and A.B = 13.5. Find the magnitude of vector |A – B| and the angle between A and B. (2011)

5. Two vectors A and B are such that |A| = 4, |B| = 6 and A.B = 8, find (2008) a) The angle between A and B b) The magnitude of |A–B|

6. Two forces of magnitude 10N and 15N are acting at a point. The magnitude of their resultant is 20N; find the angle between them. (2004)

7. Two vectors of magnitude 10 unit and 15 units are acting at a point the magnitude of their resultant are 20 units: find the angle q between them. (1993)

8. Two tugboats are towing a ship. Each exerts a force of 6000N, and the angle between the two ropes is 600. Calculate the resultant force on the ship.

26 | P a g e

9. Two vectors 10 cm and 8 cm long form an angle of, a) 600 b) 900 c) 1200 Find the magnitude of difference and the angle with respect to the larger vector.

10. Find the angles α, β, and γ which the vector A = 3i – 6j + 2k makes with the positive x, y and z axis respectively.

11. If p = 2i – k and q = 5 j Find: a) |p| b) |q| c) p + q d) q – p e) p . q f) angle between p and q. g) projection of p into q h) p x q i) unit vector perpendicular to the plane of p and q.

12. Given that: A = i + 2j + 3k and B = 2i + 4j – k, find: (1997) a) |3A – B| b) AxB c) the angle Between A and B.