Represent graphically a displacement of 40 km, 30° east of north.

Classify the following measures as scalars and vectors.

(i) 10 kg                       

ii) 2 metres north-west                         

(iii) 40°                     

(iv) 40 watt                  

(v) 10^–19 coulomb                                    

(vi) 20

Classify the following measures as scalars and vectors.

(i) 10 kg                       

(ii) 2 metres north-west                         

(iii) 40°                     

(iv) 40 watt                  

(v) 10^–19 coulomb                                    

(vi) 20

Classify the following as scalar and vector quantities.

(i) time period          

(ii) distance      

(iii) force    

(iv) velocity      

(v) work done

In Figure, identify the following vectors.

                                                          

(i) Coinitial                       

(ii) Equal                       

(iii) Collinear but not equal

In Figure, identify the following vectors.

                                                          

(i) Coinitial                       

(ii) Equal                       

(iii) Collinear but not equal

Answer the following as true or false.

(i) a and –a are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Answer the following as true or false.

(i) a and –a are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Compute the magnitude of the following vectors:

a = i + j + k,

b = 2i – 7j – 3k,

c =  + 3  –

Compute the magnitude of the following vectors:

a = i + j + k, b = 2i – 7j – 3k, c =  +  –

Write two different vectors having same magnitude.

Write two different vectors having same magnitude.

Write two different vectors having same direction.

Write two different vectors having same direction.

Find the values of x and y so that the vectors 2i + 3j and xi + yj are equal.

Find the values of x and y so that the vectors 2i + 3j and xi + yj are equal.

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Find the sum of the vectors a = i – 2j + k, b = -2i + 4j + 5k and c = i – 6j – 7k

Find the sum of the vectors a = i – 2j + k, b = -2i + 4j + 5k and c = i – 6j – 7k

Find the unit vector in the direction of the vector a = i + j + 2k.

Find the unit vector in the direction of the vector a = i + j + 2k.

Find the unit vector in the direction of vector PQ, where P and Q are the points (1, 2, 3) and  (4, 5, 6) respectively.

Find the unit vector in the direction of vector PQ, where P and Q are the points (1, 2, 3) and  (4, 5, 6) respectively.

For given vectors, a = 2i – j + 2k and b = -i + j - k, find the unit vector in the direction of the vector a + b.

For given vectors, a = 2i – j + 2k and b = -i + j - k, find the unit vector in the direction of the vector a + b.

Find a vector in the direction of vector 5i – j + 2k which has magnitude 8 units.

Find a vector in the direction of vector 5i – j + 2k which has magnitude 8 units.

Show that the vectors 2i – 3j + 4k and -4i + 6j – 8k are collinear.

Show that the vectors 2i – 3j + 4k and -4i + 6j – 8k are collinear.

Find the direction cosines of the vector i + 2j + 3k

Find the direction cosines of the vector i + 2j + 3k

Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B.

Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B.

Show that the vector i + j + k is equally inclined to the axes OX, OY and OZ.

Show that the vector i + j + k is equally inclined to the axes OX, OY and OZ.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are

i + 2j – k and –i + j + k respectively, in the ratio 2 : 1                                        

(i) internally                                    

(ii) externally

Find the position vector of a point R which divides the line joining two points P and Q

whose position vectors are i + 2j – k and –i + j + k respectively, in the ratio 2 : 1                                        

(i) internally                                    

(ii) externally

Find the position vector of the midpoint of the vector joining the points P(2, 3, 4) and Q (4, 1, – 2).

Find the position vector of the midpoint of the vector joining the points P(2, 3, 4) and Q (4, 1, – 2).

Show that the points A, B and C with position vectors,

a = 3i – 4j – 4k, b = 2i – j + k and c = i – 3j – 5k

respectively form the vertices of a right angled triangle.

Show that the points A, B and C with position vectors,

a = 3i – 4j – 4k, b = 2i – j + k and c = i – 3j – 5k

respectively form the vertices of a right angled triangle.

In triangle ABC which of the following is not true:

A. AB + BC + CA = 0

B. AB + BC - AC = 0

C. AB + BC - CA = 0

D. AB - CB + CA = 0

In triangle ABC which of the following is not true:

A. AB + BC + CA = 0

B. AB + BC - AC = 0

C. AB + BC - CA = 0

D. AB - CB + CA = 0

 

 

If a and b are two collinear vectors, then which of the following are incorrect:

A. b = λa, for some scalar λ

B. a = ±b

C. the respective components of a and b are proportional

D. both the vectors a and b have same direction, but different magnitudes

If a and b are two collinear vectors, then which of the following are incorrect:

A. b = λa, for some scalar λ

B. a = ±b

C. the respective components of a and b are proportional

D. both the vectors a and b have same direction, but different magnitudes

Find the angle between two vectors a and b with magnitudes √3 and 2, respectively having a.b = √6

Find the angle between two vectors a and b with magnitudes √3 and 2, respectively having a.b = √6

Find the angle between the vectors i – 2j +3 k and 3i – 2j + k

Find the angle between the vectors i – 2j +3 k and 3i – 2j + k

Find the projection of the vector i - j on the vector i + j.

Find the projection of the vector i - j on the vector i + j.

Find the projection of the vector i + 3j + 7k on the vector 7i - j + 8k.

Find the projection of the vector i + 3j + 7k on the vector 7i - j + 8k.

Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.

Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.

Find |a| and |b| if (a + b).(a - b) = 8 and |a| = 8|b|

Find |a| and |b| if (a + b).(a - b) = 8 and |a| = 8|b|

Evaluate the product: (3a – 5b).(2a + 7b)

Evaluate the product: (3a – 5b).(2a + 7b)

Find the magnitude of two vectors a and b, having the same magnitude and such that the angle

between them is 60° and their scalar product is .

Find the magnitude of two vectors a and b, having the same magnitude and

such that the angle between them is 60° and their scalar product is .

Find |x|, if for a unit vector a, (x - a)(x + a) = 12

Find |x|, if for a unit vector a, (x - a)(x + a) = 12

If a = 2i + 2j + 3k, b = -i + 2j + k and c = 3i + j are such that a + λb is perpendicular to c, then find the value of λ.

If a = 2i + 2j + 3k, b = -i + 2j + k and c = 3i + j are such that a + λb is

perpendicular to c, then find the value of λ.

Show that: |a|b + |b|a is perpendicular to |a|b - |b|a, for any two non-zero vectors a and b.

Show that: |a|b + |b|a is perpendicular to |a|b - |b|a, for any two non-zero vectors a and b.

If a.a = 0 and a.b = 0 then what can be concluded about the vector b?

If a.a = 0 and a.b = 0 then what can be concluded about the vector b?

If a, b, c are unit vectors such that a + b + c = 0, find the value of a ⋅ b + b ⋅ c + c ⋅ a

If a, b, c are unit vectors such that a + b + c = 0, find the value of a ⋅ b + b ⋅ c + c ⋅ a

If either vector a = 0 or b = 0, then a.b = 0. But the converse need not be true. Justify your answer with an example.

If either vector a = 0 or b = 0, then a.b = 0. But the converse need not be true. Justify your answer with an example.

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ ABC. 

[∠ ABC is the angle between the vectors BA and BC]

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ ABC. 

[∠ ABC is the angle between the vectors BA and BC]

Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Show that the vectors 2i – j + k, i – 3j – 5k and 3i – 4j – 4k form the vertices of a right angled triangle.

Show that the vectors 2i – j + k, i – 3j – 5k and 3i – 4j – 4k form the vertices of a right angled triangle.

If is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λa is unit vector if

(A) λ = 1                  

(B) λ = –1                   

(C) a = |λ|             

(D) a =

If is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λa is unit vector if

(A) λ = 1                  

(B) λ = –1                   

(C) a = |λ|             

(D) a =

Find: |a * b| if a = i – 7j + 7k and b = 3i – 2j + 2k

Find: |a * b| if a = i – 7j + 7k and b = 3i – 2j + 2k

Find a unit vector perpendicular to each of the vector a + b and a - b, where a = 3i + 2j + 2k and  b = i + 2k – 2k.

Find a unit vector perpendicular to each of the vector a + b and a - b, where a = 3i + 2j + 2k and  b = i + 2k – 2k.

If a unit vector makes an angle  with i,  with j and an acute angle θ with k, then find θ and hence, the compounds of a.

If a unit vector makes an angle  with i,  with j and an acute angle θ with k,

then find θ and hence, the compounds of a.

Show that (a - b) * (a + b) = 2(a * b)

Show that (a - b) * (a + b) = 2(a * b)

Find λ and μ if (2i + 6j + 27k) * (i + λj + μk) = 0          

Find λ and μ if (2i + 6j + 27k) * (i + λj + μk) = 0          

Given that a.b = 0 and a * b = 0. What can you conclude about the vectors a and b?

Given that a.b = 0 and a * b = 0. What can you conclude about the vectors a and b?

Let the vectors a, b, c given as a₁i + a₂j + a₃k, b₁i + b₂j + b₃k, c₁i + c₂j + c₃k, then show that: a * (b + c) = a * b + a * c

Let the vectors a, b, c given as a₁i + a₂j + a₃k, b₁i + b₂j + b₃k, c₁i + c₂j + c₃k, then show that: a * (b + c) = a * b + a * c

If either a = 0 or b = 0, then a * b = 0. Is the converse true?                                                       

Justify your answer with an example.

If either a = 0 or b = 0, then a * b = 0. Is the converse true?                                                       

Justify your answer with an example.

Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Find the area of the parallelogram whose adjacent sides are determined by the vector a = i – j + 3k and b = 2i – 7j + k

Find the area of the parallelogram whose adjacent sides are determined by the vector a = i – j + 3k and b = 2i – 7j + k

Let the vectors and be such that |a| = 3 and |b| = , the a * b is a unit vector, if the angle between a and b is                                                                                                                                   

(A)                                

(B)                                  

(C)                                 

(D)

Let the vectors and be such that |a| = 3 and |b| = , the a * b is a unit vector, if the angle between a and b is                                                                                                                                   

(A)                                

(B)                                  

(C)                                 

(D)

Area of a rectangle having vertices A, B, C, and D with position vectors –i +  + 4k, i +  + 4k, i –  + 4k and –i –  + 4k respectively is                                                                                           

(A)                                   

(B) 1                               

(C) 2                                    

(D) 4

Area of a rectangle having vertices A, B, C, and D with position vectors –i +  + 4k, i +  + 4k, i –  + 4k and –i –  + 4k respectively is                                                                                           

(A)                                   

(B) 1                               

(C) 2                                    

(D) 4

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Find the scalar components and magnitude of the vector joining the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).

Find the scalar components and magnitude of the vector joining the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops.

Determine the girl’s displacement from her initial point of departure.

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops.

Determine the girl’s displacement from her initial point of departure.

If a = b + c, then is it true that |a| = |b| + |c|? Justify your answer.

If a = b + c, then is it true that |a| = |b| + |c|? Justify your answer.

Find the value of x for which x(i + j + k) is a unit vector.

Find the value of x for which x(i + j + k) is a unit vector.

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a = 2i + 3j – k and b = i – 2j + k.

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a = 2i + 3j – k and b = i – 2j + k.

If a = i + j + k, b =2 i - j + 3k and c = i - 2j + k find a unit vector parallel to the vector 2a – b + 3c.

If a = i + j + k, b =2 i - j + 3k and c = i - 2j + k find a unit vector parallel to the vector 2a – b + 3c.

Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a + b) and (a – 3b)

externally in the ratio 1: 2. Also, show that P is the mid-point of the line segment RQ.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors

are (2a + b) and (a – 3b) externally in the ratio 1: 2.

Also, show that P is the mid-point of the line segment RQ.

The two adjacent sides of a parallelogram are 2i – 4j + 5k and i – 2j – 3k.

Find the unit vector parallel to its diagonal. Also, find its area.

The two adjacent sides of a parallelogram are 2i – 4j + 5k and i – 2j – 3k.

Find the unit vector parallel to its diagonal. Also, find its area.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ,  and .

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ,  and .

Let a = i + 4j + 2k, b = 3i – 2j + 7k and c = 2i – j + 4k.

Find a vector d which is perpendicular to both a and b, and c.d = 15.

Let a = i + 4j + 2k, b = 3i – 2j + 7k and c = 2i – j + 4k. Find a vector d which is perpendicular to both a and b, and c.d = 15.

The scalar product of the vector i + j + k with a unit vector along the sum of vectors

2i + 4j – 5k and λi + 2j + 3k is equal to one. Find the value of λ.

The scalar product of the vector i + j + k with a unit vector along the sum of vectors

2i + 4j – 5k and λi + 2j + 3k is equal to one. Find the value of λ.

If a, b and c are mutually perpendicular vectors of equal magnitudes,

show that the vector a + b + c is equally inclined to a, b and c.

If a, b and c are mutually perpendicular vectors of equal magnitudes,

show that the vector a + b + c is equally inclined to a, b and c.

Prove that (a + b).(a + b) = |a|² + |b|² if and only if a, b are perpendicular, given a ≠ 0, b ≠ 0.

Prove that (a + b).(a + b) = |a|² + |b|² if and only if a, b are perpendicular, given a ≠ 0, b ≠ 0.

If θ is the angle between two vectors a and b, then a.b ≥ 0 only when

(A) 0 < θ <               

(B) 0 ≤ θ ≤               

(C) 0 < θ < π             

(D) 0 ≤ θ ≤ π

If θ is the angle between two vectors a and b, then a.b ≥ 0 only when

(A) 0 < θ <               

(B) 0 ≤ θ ≤               

(C) 0 < θ < π             

(D) 0 ≤ θ ≤ π

Let a and b be two unit vectors and θ is the angle between them. Then a + b is a unit vector if

(A) θ =                          

(B) θ =                          

(C) θ =                          

(D) θ =

Let a and b be two unit vectors and θ is the angle between them. Then a + b is a unit vector if

(A) θ =                          

(B) θ =                          

(C) θ =                          

(D) θ =

The value of is i.(j * k) + j.(i * k) + k.(i * j) is

(A) 0                                 

(B) –1                              

(C) 1                                

(D) 3

The value of is i.(j * k) + j.(i * k) + k.(i * j) is

(A) 0                                 

(B) –1                              

(C) 1                                

(D) 3

If θ be the angle between two vectors a and b, then |a.b| = |a * b|, when θ is equal to

(A) 0                              

(B)                    

(C)                      

(D) π

If θ be the angle between two vectors a and b, then |a.b| = |a * b|, when θ is equal to

(A) 0                              

(B)                    

(C)                      

(D) π