CLASS XII MATH VECTOR DPP
CLASS XII - VECTOR
1. If a and b are unit vectors, then what is the angle between a and b for
√3 a - b to be a unit vector?
(a) 30° (b) 45° (c) 60° (d) 90°
2. The value of λ, for which vectors 2i + j + 3k and i - λj + 4k are
orthogonal is ______.
3. Find the angle between the vectors a = i - j + k and b = i + j - k.
4. If |a| = √3, |b| = 2 and angle between a and b is 60°, find a . b.
5. Find the projection of a on b if a•b = 8 and b = 2i + 6j + 3k.
6. If p is a unit vector and (x - p) • (x + p) = 80, then find |x|.
7. If a • a = 0 and a • b = 0, then what can be concluded about the vector
b? [Foreign 2011]
8. Find λ, when the projection of a = λi + j + 4k on b = 2i + 6j + 3k is 4
units. [Delhi 2012]
9. Show that |a + b|² = |a|² + |b|² , if a and b are along adjacent sides of a
rectangle.
10. If a is a unit vector and (x - a) • (x + a) = 8, find |x|. [NCERT]
11. Find the magnitude of each of the two vectors a and b, having the same
magnitude such that the angle between them is 60° and their scalar
product 9/2 [CBSE 2018]
12. If two vectors a and b are such that |a| = 2, |b| = 1 and a • b = 1, then
find the value of
(3a - 5b) • (2a + 7b). [Delhi 2011]
13. Find the projection of b + c on a, where
a = 2i - 2j + k, b = i + 2j - 2k and c = 2i - j + 4k.
14. If a = 7i + j - 4k and b = 2i + 6j + 3k, then find the projection of a on b.
[Delhi 2015]
15. If |a| = 2, |b| = √3 and a . b = √3, find the angle between a and b.
16. If a and b are two unit vectors and θ is the angle between them, then
show that 1/2 (a - b)2 = 1 - cos θ.
17. If (a + b) . (a - b) = 12 and |a| = 2, find |a| and |b|.
✤ Long Answer I / Long Answer II Type
18. If a = i + 2j - 3k, b = 3i - j + 2k, show that (a + b) and (a - b) are
perpendicular to each other.
19. The scalar product of the vector i + j + k with the unit vector along
the sum of vectors 2i + 4j - 5k and λi + 2j + 3k is equal to one. Find
the value of λ.
20. If a, b are any two vectors, then give the geometrical interpretation of
the relation |a + b| = |a - b|
21. Show that each of the given three vectors is a unit vector
1/7(2i + 3j + 6k), 1/7(3i - 6j + 2k), 1/7(6i + 2j - 3k).
Also, show that they are mutually perpendicular to each other.
[NCERT]
22. If the vertices A, B, C of a Δ ABC have position vectors
(1, 2, 3), (-1, 0, 0) and (0, 1, 2) respectively, what is the magnitude of
∠ABC?
23. If a+b+c = 0 and |a| = 3, |b| = 5 and |c| = 7, then find the angle
between a and b. [Delhi 2014]
24. If a,b,c are mutually perpendicular vectors of equal magnitudes, show
that the vector a+b+c is equally inclined to a, b and c. Also, find the
angle which a+b+c makes with a or b or c.
[Delhi 2017]
25. If a = 2i - j - 2k and b = 7i + 2j - 3k, then express b in the form of b =
b1 + b2, where b1 is parallel to a and b2 is perpendicular to a.
[AI 2017]
26. If vectors 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.
[NCERT, Foreign 2011]
27. Dot product of a vector with vectors i - j + k, 2i + j - 3k and i + j + k
are respectively 4, 0 and 2. Find the vector. [Delhi 2013(C)]
29. If a and b are two vectors such that |a + b| = |a|, then prove that vector 2a + b is perpendicular to vector b.
✤ PRACTICE QUESTIONS
Very Short (Objective Type) / Short Answer Type
1. For any non zero vector a,
a . ( a . i ) + (a . j) + (a . k). State true or false.
2. If |a + b| = |a - b|, then a and b are perpendicular.
State true or false.
3. Write the projection of the vector i - j on the vector i + j.
[NCERT, AI 2011]
4. If a is a unit vector and (2x - 3a).(2x + 3a) = 91, then write the value
of |x|. [Delhi 2013(C)]
5. Find the projection of the vector i + 3j + 7k on the vector 2i - 3j + 6k.
[Delhi 2014]
6. For what value of λ are the vectors a = 2i + 2j + k and b = -i - 2j + 3k
perpendicular to each other? [Delhi 2013(C)]
7. If a and b are two unit vectors such that a + b is also a unit vector,
then find the angle between a and b. [Delhi 2014]
8. If a, b and c are mutually perpendicular unit vectors, then find the
value of |2a + b + c|.
9. Find the length of the sum of the three mutually perpendicular unit
vectors.
10. If (a + b) is perpendicular to b and a is perpendicular to (2b + a) then
show that a² = 2b².
Long Answer I / Long Answer II Type
11. If a = 3i + 4j + 5k and b = 2i + j - 4k, then express b in the form b =
b₁ + b₂, where b₁ is parallel to a and b₂ is perpendicular to a.
[Foreign 2012]
12. If a = i - j + 7k and b = 5i - j + λk, then find the value of λ, so that a +
b and a - b are perpendicular vectors. [AI 2013]
13. If a, b and c are three mutually perpendicular vectors of equal
magnitude, prove that the angle which (a + b + c) makes with any of
the vectors a, b or c is cos⁻¹ ( 1 / √3 ). [DSE]
14. If a and b are two unit vectors and θ is the angle between them, then
show that sin (θ/2) = (1/2)|a - b|. [HOTS]
15. The scalar product of the vector a - i + j + k with a unit vector along
the sum of the vectors b = 2i + 4j - 5k and c = λi + 2j + 3k is equal to
one. Find the value of λ, and hence find the unit vector along b + c.
[AI 2014]
16. If a, b, c are three vectors such that |a| = 5, |b| = 12 and |c| = 13, and
a + b + c = 0, find the value of a . b + b . c + c . a. [Delhi 2012]
17. Show that the angle between any two diagonals of a cube is
cos⁻¹(1 / 3).
✤ CROSS PRODUCT OR VECTOR PRODUCT
1. Vector or cross product of two vectors a and b, denoted by a x b, is
given by |a x b| = |a||b| sin θ, where θ is angle between a and b and ǹ
is a unit vector perpendicular to a and b and direction is such that
a, b and ǹ form a right hand system.
2. |a x b| is a vector quantity, whose magnitude is, |a x b| = |a||b| sin θ.
3. If θ is angle between a and b, then sin θ = |a x b| / |a||b|
4. For a, a x a = 0.
5. For vectors a and b, a x b = -b x a, i.e. cross product of two vectors is
not commutative.
6. For a, b and c, a x (b x c) ≠ (a x b) x c, in general.
7. Distributive property: For vectors a, b and c, a x (b + c) = a x b + a x c
8. For vectors a and b, a x b = 0 ⇔ a = 0, b = 0 or a || b.
9. If we want to show that two non-zero vectors a and b are parallel,
then we should show that a x b = 0.
10. Geometrically, |a x b| represents the area of a parallelogram whose
adjacent sides are along a and b.
11. Area of a triangle whose sides are along a and b is given by 1/2 |a x b|.
12. Area of a parallelogram whose diagonals are along d1 and d2 is given
by 1/2|d1 x d2|.
13. If i, j, k are vectors along x, y and z-axes respectively, then
i x i = 0, j x j = 0, k x k = 0, i x j = k;
j x k = i; k x i = j.
14. For a scalar λ, λ(a x b) = (λa x b) = (a x λb), where a and b are given
vectors.
15. If a = a1i + b1j + c1k; b = a2i + b2j + c2k, then a x b =
| i j k |
|a1 b1 c1|
|a2 b2 c2|
16. Unit vector ǹ perpendicular to a and b is ǹ = a x b / |a x b|
(i) If we have to find unit vectors perpendicular to a and b, then we
take ±ǹ.
(ii) If we have to find a vector of magnitude k units and
perpendicular to vectors a and b, then we find k ǹ,
i.e. k (a x b / |a x b|)
(iii) If we have to find vectors of magnitude k units and
perpendicular to vectors a and b, then we find
± k ǹ, i.e. ±k (a x b / |a x b|)
✤ SOLVED EXAMPLES
Very Short (Objective Type) / Short Answer Type
1. If for non zero vectors a and b, |a x b| is a unit vector and
|a| = |b| = √2, then angle θ between vectors a and b is
(a) π/2 (b) π/3 (c) π/6 (d) -π/2
2. The area of a parallelogram whose one diagonal is 2i + j - 2k and one
side is 3i + j - k is
(a) i - 4j - k (b) 3√2 sq units
(c) 6√2 sq units (d) 6 sq units
3. Find the angle between two vectors a and b with magnitudes 1 and 2
respectively and when |a x b| = √3.
4. Find λ, if (2i + 6j + 14k) x (i - λj + 7k) = 0.
5. If a x b = b x c and a x c = b x a, prove that a - b is parallel to b - c,
provided a = d and b = c.
6. Show that the area of the parallelogram having diagonals (3i - j - 2k)
and (i - 3j - 4k) is 5√3 sq units.
7. Write the value of (i x j).k + j. [AI 2012]
8. Write a unit vector perpendicular to both the vectors
a = i + j + k and b = i + j. [AI 2015]
9. If a = i - j - k and b = j - k and a x c = b and a.c = 3.
10. Find a unit vector perpendicular to each of the vectors a + b and a - b,
where a = 3i + 2j + 2k and b = i + 2j - 2k. [NCERT: Delhi 2001]
11. a, b, c are unit vectors, suppose a ⋅ b = a ⋅ c = 0 and angle between b
and c is . Prove that → a = ±2(b × c).
[CBSE Sample paper 2016]
12. If for three non-zero vectors a, b and c, a ⋅ b = a ⋅ c and a × b = a × c,
then show that b = c.
13. Using vectors, find the area of the triangle with vertices A(1, 1, 2),
B(2, 3, 5) and C(1, 5, 5). [AI 2011]
14. Find a vector of magnitude 6, perpendicular to each of the vectors
a + b and a - b, where a = i + j + k and b = i + 2j + 3k.
[Foreign 2013]
15. If r = xi + yj + zk, find (r × i) . (r × j) + xy.
16. Show that the points A, B, C with position vectors
2i - j + k, i - 3j - 5k and 3i - 4j - 4k
respectively, are the vertices of a right-angled triangle. Hence find the
area of the triangle. [AI 2017]
17. Let a = 4i + 5j - k, b = i - 4j + 5k and c = 3i + j - k.
Find a vector d which is perpendicular to both c and b and d . a = 21.
✤ Very Short (Objective Type) / Short Answer Type
1. If |a| = 8, |b| = 3 and |a · b| = 12√3 then the value of |a x b| is
(a) 12 (b) 12√3 (c) 6 (d) 4√3
2. Vectors a and b are such that |a| = √3, |b| = 2/3 and (a x b) is a unit
vector. Write the angle between a and b.
3. Find the area of a parallelogram whose adjacent sides are represented
by the vectors 2i - 3k and 4j + 2k.
[Foreign 2015]
4. If a and b are two vectors such that |a| = |a x b|. Then what is the
angle between a and b?
5. Write the value of the area of the parallelogram determined by the
vectors 2i and 3j.
[Foreign 2012]
✤ Long Answer I / Long Answer II Type
6. 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 = 18.
7. If a x b = b x c ≠ 0, show that a + c = mb, where m is a scalar.
8. Find a vector whose magnitude is 3 units and which is perpendicular
to the following two vectors:
a = 3i + j - 4k; b = 6i + 5j - 2k.
SCALAR TRIPLE PRODUCT
1. Scalar triple product: a.(b x c) or (a x b).c.
For vectors a, b, c, scalar triple product is given by a.(b x c) or (a x b).c.
Scalar triple product is also denoted by [a b c].
2. For vectors a = a₁i + b₁j + c₁k, b = a₂i + b₂j + c₂k and
c = a₃i + b₃j + c₃k
[a b c] = | a₁ b₁ c₁ |
| a₂ b₂ c₂ |
| a₃ b₃ c₃ |
3. [a b c] = - [b c a] = - [c a b].
4. [a b c] = - [c a b].
5. [a b b] = 0; [a a b] = 0.
6. If [a b c] = 0, then vectors a, b, c are coplanar and vice versa.
7. If vectors a = a₁i + b₁j + c₁k, b = a₂i + b₂j + c₂k and
c = a₃i + b₃j + c₃k are coplanar,
then | a₁ b₁ c₁ | = 0
| a₂ b₂ c₂ |
| a₃ b₃ c₃ |
8. [a b c] geometrically represents volume of a parallelepiped whose
adjacent sides are along a, b, c.
9. a, b, c are three non zero vectors such that a x b = c, b x c = a. Prove
that a, b, c are mutually at right angles and |b| = 1, |c| = |a|. [HOTS]
10. If a = i + j + k and b = j - k, find a vector c such that a x c = b and
a · c = 3. [NCERT Exemplar]
11. If a + b + c = 0, show that a x b = b x c = c x a. Also, interpret the
result geometrically. [NCERT Exemplar]
12. Find a unit vector perpendicular to both of the vectors a + b and a - b
where a = i + j + k, b = i + 2j + 3k.
[Foreign 2014]
13. If a, b, c are position vectors of vertices A, B, C of a triangle ABC, show
that area of a triangle is 1/2 | a x b + b x c + c x a |.
14. Find the area of triangle OAB, where OA = 3i - j + k and OB = -2i + j –
3k.
15. If a x b = a x c ≠ 0, show that b = c + λa, for some real number λ.
16. For any two vectors a and b, prove that (a x b)² = |a|² |b|² - (a.b)².
17. Define vector product a x b of two vectors a and b. If |a| = 2, |b| = 5 and |a x b| = 8, find the value of a · b. [HOTS]
1. If for three non zero vectors a, b, c, |a b c| = 0, then vectors a, b, c are
_____.
2. Find λ, if the vectors a = i + 3j + k, b = 2i - j – k and c = λj + 3k are
coplanar. [Delhi 2015]
3. Find λ, so that the vectors 2i - j + k, i + 2j - 3k and 3i + λj + 5k are
coplanar.
✤ Long Answer I / Long Answer II Type
4. Find the volume of a parallelepiped whose continuous edges are
represented by vectors a = 2i - 3j + k, b = i - j + 2k and c = 2i + j - k.
5. Let a, b, c, be three non-zero vectors. If a.(b x c) = 0 where b, c are not
parallel, then prove that a = λb + μc, where λ and μ are scalar
constants.
6. Find x, such that the points A(3, 2, 1), B(4, x, 5), C(4, 2, -2) and
D(6, 5, -1) are coplanar.
7. Show that the vectors a, b, c are coplanar/non-coplanar if a + b, b + c,
c + a are coplanar / non-coplanar. [NCERT: Delhi 2014]
8. Show that the four points A, B, C and D with position vectors
4i + 5j + k, -j - k, 3i + 9j + 4k and 4(-i + j + k) respectively are
coplanar. [AI 2014]
9. If a = i − 2j + 3k, b = 2i + 3j − 5 k, then find a × b. Verify that a and
a × b are perpendicular to each other.
✤ PRACTICE QUESTIONS
❈ Short (Objective Type) / Short Answer Type
1) If vectors i + j − 3k, 2i + j − λk, 5i + 2j + 3k are coplanar, then value
of λ is
(a) 4 (b) 0 (c) −3 (d) 2
2) Find a.(b × c), if a = 2i + j + 3k, b = −i + 2j + k and c = 3i + j + 2k.
[AI 2014]
3) If a = i − j + k and b = 3i + 2j + 4k, find [a b].
4) Position vectors of points A and B are a + b and 2a − b. Then AB equal
to
(a) 3a (b) −a + 2b (c) a − 2b (d) none of these
5) Given vector a, then −2a is a vector whose
(a) magnitude is twice that of a and direction is same as that of a
(b) magnitude is twice that of a and direction is opposite to that of a
(c) magnitude is same as that of a and direction is opposite to that
of a
(d) none of these
✤ Long Answer I / Long Answer II Type
6) If any three vectors a, b, c are coplanar, show that the vectors a + b, b
+ c and c + a are also coplanar.
[DoE; Delhi 2016]
7) Using vectors show that the points A(−1, 4, −3),
B(3, 2, −5), C(−3, 8, −5) and D(−3, 2, 1) are coplanar.
8) Let a = i + j + k, b = i and c = c1i + c2j + c3k, If c2= −1 and c₁ = 1,
show that no value of c3 can make a, b, c coplanar.
[NCERT; Delhi 2017]
9) Show that the vectors a, b, c are coplanar if and only if a + b, b + c and
c + a are coplanar. [Foreign 2014]
* * * * * * * * * * * * * * *
