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Solutions

Overview

Step-by-step NCERT solutions for Vector Algebra (Chapter 10, CBSE Class 12 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Vector Algebra textbook chapter.

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What these solutions cover

All 73 questions in Vector Algebra are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 10.1

  1. Represent graphically a displacement of 40 km, 30° east of north.
  2. Classify the following measures as scalars and vectors.
    • (i) 10 kg
    • (ii) 2 meters north-west
    • (iii) 40°
    • (iv) 40 watt
    • (v) 10^(-19) coulomb
    • (vi) 20 m/s²
  3. Classify the following as scalar and vector quantities.
    • (i) time period
    • (ii) distance
    • (iii) force
    • (iv) velocity
    • (v) work done
  4. In Fig 10.6 (a square), identify the following vectors.
    • (i) Coinitial
    • (ii) Equal
    • (iii) Collinear but not equal
  5. 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.

Exercise 10.2

  1. Compute the magnitude of the following vectors: a = î + ĵ + k̂; b = 2î − 7ĵ − 3k̂; c = (1/√3)î + (1/√3)ĵ − (1/√3)k̂
  2. Write two different vectors having same magnitude.
  3. Write two different vectors having same direction.
  4. Find the values of x and y so that the vectors 2î + 3ĵ and xî + yĵ are equal.
  5. Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).
  6. Find the sum of the vectors a = î − 2ĵ + k̂, b = −2î + 4ĵ + 5k̂ and c = î − 6ĵ − 7k̂.
  7. Find the unit vector in the direction of the vector a = î + ĵ + 2k̂.
  8. 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.
  9. For given vectors, a = 2î − ĵ + 2k̂ and b = −î + ĵ − k̂, find the unit vector in the direction of the vector a + b.
  10. Find a vector in the direction of vector 5î − ĵ + 2k̂ which has magnitude 8 units.
  11. Show that the vectors 2î − 3ĵ + 4k̂ and −4î + 6ĵ − 8k̂ are collinear.
  12. Find the direction cosines of the vector î + 2ĵ + 3k̂.
  13. Find the direction cosines of the vector joining the points A(1, 2, −3) and B(−1, −2, 1), directed from A to B.
  14. Show that the vector î + ĵ + k̂ is equally inclined to the axes OX, OY and OZ.
  15. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are î + 2ĵ − k̂ and −î + ĵ + k̂ respectively, in the ratio 2 : 1
    • (i) internally
    • (ii) externally
  16. Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, −2).
  17. Show that the points A, B and C with position vectors a = 3î − 4ĵ − 4k̂, b = 2î − ĵ + k̂ and c = î − 3ĵ − 5k̂, respectively form the vertices of a right angled triangle.
  18. In triangle ABC (Fig 10.18), 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
  19. 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 not proportional (D) both the vectors a and b have same direction, but different magnitudes.

Exercise 10.3

  1. Find the angle between two vectors a and b with magnitudes √3 and 2, respectively having a · b = √6.
  2. Find the angle between the vectors î − 2ĵ + 3k̂ and 3î − 2ĵ + k̂.
  3. Find the projection of the vector î − ĵ on the vector î + ĵ.
  4. Find the projection of the vector î + 3ĵ + 7k̂ on the vector 7î − ĵ + 8k̂.
  5. Show that each of the given three vectors is a unit vector: (1/7)(2î + 3ĵ + 6k̂), (1/7)(3î − 6ĵ + 2k̂), (1/7)(6î + 2ĵ − 3k̂) Also, show that they are mutually perpendicular to each other.
  6. Find |a|² and |b|², if (a + b) · (a − b) = 8 and |a| = 8|b|.
  7. Evaluate the product (3a − 5b) · (2a + 7b).
  8. 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 1/2.
  9. Find |x|, if for a unit vector â, (x − â) · (x + â) = 12.
  10. If a = 2î + 2ĵ − 3k̂, b = 3î − ĵ + 2k̂ are such that a + λb is perpendicular to b, then find the value of λ.
  11. Show that |a|b + |b|a is perpendicular to |a|b − |b|a, for any two nonzero vectors a and b.
  12. If a · a = 0 and a · b = 0, then what can be concluded about the vector b?
  13. If a, b, c are unit vectors such that a + b + c = 0, find the value of a · b + b · c + c · a.
  14. 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.
  15. 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].
  16. Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, −1) are collinear.
  17. Show that the vectors 2î − ĵ + k̂, î − 3ĵ − 5k̂ and 3î − 4ĵ − 4k̂ form the vertices of a right angled triangle.
  18. If a 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 = 1/|λ|

Exercise 10.4

  1. Find |a × b|, if a = î − 7ĵ + 7k̂ and b = 3î − 2ĵ + 2k̂.
  2. Find a unit vector perpendicular to each of the vectors (a + b) and (a − b), where a = 3î + 2ĵ + 2k̂ and b = î + 2ĵ − 2k̂.
  3. If a unit vector â makes angles π/3 with î, π/4 with ĵ and an acute angle θ with k̂, then find θ and hence, the components of â.
  4. Show that (a − b) × (a + b) = 2(a × b).
  5. Find λ and μ if (2î + 6ĵ + 27k̂) × (î + λĵ + μk̂) = 0.
  6. Given that a · b = 0 and a × b = 0. What can you conclude about the vectors a and b?
  7. Let the vectors a, b, c be given as a₁î + a₂ĵ + a₃k̂, b₁î + b₂ĵ + b₃k̂, c₁î + c₂ĵ + c₃k̂. Then show that a × (b + c) = a × b + a × c.
  8. If either a = 0 or b = 0, then a × b = 0. Is the converse true? Justify your answer with an example.
  9. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
  10. Find the area of the parallelogram whose adjacent sides are determined by the vectors a = î − ĵ + 3k̂ and b = 2î − 7ĵ + k̂.
  11. Let the vectors a and b be such that |a| = 3 and |b| = √2/3, then a × b is a unit vector, if the angle between a and b is (A) π/6 (B) π/4 (C) π/3 (D) π/2
  12. Area of a rectangle having vertices A, B, C and D with position vectors −î + (1/2)ĵ + 4k̂, î + (1/2)ĵ + 4k̂, î − (1/2)ĵ + 4k̂ and −î − (1/2)ĵ + 4k̂, respectively is (A) 1/2 (B) 1 (C) 2 (D) 4

Miscellaneous Exercise

  1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
  2. Find the scalar components and magnitude of the vector joining the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).
  3. 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.
  4. If ā = b̄ + c̄, then is it true that |ā| = |b̄| + |c̄|? Justify your answer.
  5. Find the value of x for which x(î + ĵ + k̂) is a unit vector.
  6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors ā = 2î + ĵ + k̂ and b̄ = î − 2ĵ − k̂.
  7. If ā = î + ĵ + k̂, b̄ = 2î − ĵ + 3k̂ and c̄ = î − 2ĵ + k̂, find a unit vector parallel to the vector 2ā − b̄ + 3c̄.
  8. 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.
  9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2ā + b̄) and (ā − 3b̄) externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
  10. The two adjacent sides of a parallelogram are 2î − 4ĵ + 5k̂ and î − 2ĵ − 3k̂. Find the unit vector parallel to its diagonal. Also, find its area.
  11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ±(1/√3, 1/√3, 1/√3).
  12. Let ā = î + 4ĵ + 2k̂, b̄ = 3î − 2ĵ + 7k̂ and c̄ = 2î − ĵ + 4k̂. Find a vector d̄ which is perpendicular to both b̄ and c̄, and ā · d̄ = 15.
  13. The scalar product of the vector î + ĵ + k̂ with a unit vector along the sum of vectors 2î + 4ĵ − 5k̂ and λî + 2ĵ + 3k̂ is equal to one. Find the value of λ.
  14. If ā, b̄, c̄ are mutually perpendicular vectors of equal magnitudes, show that the vector ā + b̄ + c̄ is equally inclined to ā, b̄ and c̄.
  15. Prove that (ā + b̄) · (ā + b̄) = |ā|² + |b̄|², if and only if ā, b̄ are perpendicular, given ā ≠ 0, b̄ ≠ 0.
  16. If θ is the angle between two vectors ā and b̄, then ā · b̄ ≥ 0 only when (A) 0 < θ < π/2 (B) 0 ≤ θ ≤ π/2 (C) 0 < θ < π (D) 0 ≤ θ ≤ π
  17. Let ā and b̄ be two unit vectors and θ is the angle between them. Then ā + b̄ is a unit vector if (A) θ = π/4 (B) θ = π/3 (C) θ = π/2 (D) θ = 2π/3
  18. The value of î · (ĵ × k̂) + ĵ · (î × k̂) + k̂ · (î × ĵ) is (A) 0 (B) −1 (C) 1 (D) 3
  19. If θ is the angle between any two vectors ā and b̄, then |ā · b̄| = |ā × b̄| when θ is equal to (A) 0 (B) π/4 (C) π/2 (D) π
Sample

Worked examples

3 questions from Vector Algebra, solved step by step.

Exercise 10.1

Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 meters north-west (iii) 40° (iv) 40 watt (v) 10^(-19) coulomb (vi) 20 m/s²

(i) 10 kg — mass has only magnitude, no direction. Scalar.

(ii) 2 meters north-west — has both magnitude (2 m) and direction (north-west). Vector.

(iii) 40° — an angle, has magnitude only. Scalar.

(iv) 40 watt — power has only magnitude. Scalar.

(v) 10^(-19) coulomb — electric charge has only magnitude. Scalar.

(vi) 20 m/s² — acceleration has both magnitude and direction. Vector.

Answer: Scalars: (i), (iii), (iv), (v). Vectors: (ii), (vi).

Exercise 10.2

Write two different vectors having same direction.

Two vectors have the same direction if one is a positive scalar multiple of the other.

For example: a = î + ĵ + k̂ and b = 2î + 2ĵ + 2k̂.

Both point in the direction (1, 1, 1) but have different magnitudes (√3 and 2√3).

Answer: a = î + ĵ + k̂ and b = 2(î + ĵ + k̂) = 2î + 2ĵ + 2k̂ have the same direction (any valid positive-scalar-multiple pair is acceptable).

Exercise 10.3

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

AB = (2−1)î + (6−2)ĵ + (3−7)k̂ = î + 4ĵ − 4k̂.

BC = (3−2)î + (10−6)ĵ + (−1−3)k̂ = î + 4ĵ − 4k̂.

Since AB = BC, both vectors are identical, meaning B is the midpoint of AC and all three points lie on the same line.

Also, AC = 2î + 8ĵ − 8k̂ = 2·AB, confirming A, B, C are collinear.

Answer: AB = BC = î + 4ĵ − 4k̂, so A, B, C are collinear.

All 73 questions in Vector Algebra are solved step by step in the Solutions PDF above.

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