Karnataka 2nd PUC Maths Chapter 10 Important Questions 2027: Vector Algebra

Vector Algebra covers scalars and vectors, types of vectors, magnitude, direction cosines, vector addition, scalar multiplication, dot product and cross product. It also includes applications such as projections and finding areas using the cross product, along with angle and perpendicularity conditions between vectors.

Students can use these Karnataka 2nd PUC Maths Chapter 10 Important Questions 2027 for focused revision and practice. Regular practice of these questions helps build accuracy and speed for both board exams and KCET.

Karnataka 2nd PUC Vector Algebra Important Questions

The following Vector Algebra Questions and Answers cover unit vectors, vector addition, section formula, dot product, perpendicularity, projections and cross product. These 60 MCQs include questions from KCET 2013 to 2026 for comprehensive practice.

1. Vector Algebra – 1 Mark Questions

These are quick questions to check if you understand the basics of vectors — like telling scalars and vectors apart, knowing how the triangle law works, spotting when vectors lie on the same line (collinear), and figuring out unit vectors from a scaled vector.

Q1. Which of the following is a vector?
(1) time 

(2) volume 

(3) force 

(4) speed
 

Answer: (3) force

Q2. In the given triangle ABC, which of the following is not true?

 (1) AB + BC − CA = 0
(2) AB + BC − AC = 0
(3) AB + BC + CA = 0
(4) AB − CB + CA = 0
 

Answer: (1) AB + BC − CA = 0

Q3. If a and b are two collinear vectors, then which of the following is incorrect?
(1) b = λa, for some λ
(2) a = ±b
(3) The respective components of a and b are not proportional
(4) Both vectors have the same direction but different magnitudes

Answer: (3) The respective components of a and b are not proportional

Q4. If a is a nonzero vector of magnitude a and λ a nonzero scalar, then λa is a unit vector if
(1) λ = 1 (2) λ = −1 (3) a = |λ| (4) a = 1/|λ|

Answer: (4) a = 1/|λ|

Q5. If i, j, k are mutually perpendicular unit vectors, the value of i·(j × k) is
(1) 0 

(2) 1 

(3) −1 

(4) 3

Answer: (2) 1

Vector Algebra – 2 Mark Questions

These questions need a bit of calculation. You might have to find the vector between two points, make a unit vector in a certain direction, use the section formula, or compare the lengths of sums and differences of unit vectors.

Q1. A vector of magnitude 5 units and in the direction of the vector 2i − 6j + 3k is
(1) (5/7)(2i − 6j + 3k)
(2) (5/7)(−2i + 6j − 3k)
(3) (1/7)(2i − 6j + 3k)
(4) (1/7)(−2i + 6j − 3k)
 

Answer: (1) (5/7)(2i − 6j + 3k)

Q2. The unit vector in the direction of the vector a = i − 2j + 2k is

(1) (1/9)i − (2/9)j + (2/9)k
(2) −(1/3)i + (2/3)j − (2/3)k
(3) −(1/9)i + (2/9)j − (2/9)k
(4) (1/3)i − (2/3)j + (2/3)k

Answer: (4) (1/3)i − (2/3)j + (2/3)k

Q3. The position vector of a point C which divides the line segment joining A and B (position vectors a and b) internally in the ratio 1:2 is

 (1) (a + 2b)/3 

(2) (2a + b)/3 

(3) (a − b)/3 

(4) (a + b)/3

Answer: (2) (2a + b)/3

Q4. The sum of two unit vectors is a unit vector. The magnitude of their difference is
(1) 0 

(2) 1 

(3) √2 

(4) √3

Answer: (4) √3

Q5. If a and b are two collinear vectors, then

(1) a × b = 0
(2) a = λb for some scalar λ
(3) a·b = |a||b|
(4) All of the above

Answer: (4) All of the above

Vector Algebra – 3/5 Mark Questions

These are longer, multi-step problems, usually taken from recent KCET papers. They cover things like finding the length of a median in a triangle, checking coplanarity when one value is unknown, and finding an unknown vector or scalar using given dot or cross product conditions.

Q1. Vectors i + j + k and i + 3j + 5k represent sides AB and AC of ∆ABC. The length of the median through A is
(1) √14 

(2) √14/2 

(3) 14 

(4) 7
Answer: (1) √14

Q2. If the vectors ai + j + k, i + bj + k and i + j + ck are coplanar (a ≠ b ≠ c ≠ 1), then abc − (a + b + c) =
 

(1) 2 

(2) −2 

(3) 0 

(4) −1
Answer: (2) −2

Q3. If a = i + 2j + k, b = i − j + 4k and c = i + j + k are such that a + λb is perpendicular to c, then λ =
(1) 1 

(2) −1 

(3) 2 

(4) −2

Answer: (2) −1

Q4. If a and b are unit vectors, the angle between a and b for √3a − b to be a unit vector is
(1) 30° 

(2) 45° 

(3) 60° 

(4) 90°
 

Answer: (1) 30°

Q5. If a = i + j + k, b = j − k, and a × c = b, a·c = 3, then c is equal to
(1) (1/3)(5i + 2j + 2k)
(2) (1/3)(2i + 5j + 2k)
(3) (1/3)(5i + 2j − k)
(4) i + j + k

Answer: (1) (1/3)(5i + 2j + 2k)

Vector Algebra – Formula-Based Questions

These questions test direct application of standard formulae — area of a figure from position vectors, the condition for λa to be a unit vector, the sum/difference-of-vectors identity, and the coplanarity determinant condition.

Q1. Area of a rectangle with vertices having position vectors −i + ½j + 4k, i + ½j + 4k, i − ½j + 4k and −i − ½j + 4k is
(1) 1/2 

(2) 1 

(3) 2 

(4) 4

Answer: (3) 2

Q2. If a is a non-zero vector of magnitude a and λ is a non-zero scalar, then λa is a unit vector if
(1) λ = |a| 

(2) λ = 1/|a| 

(3) λ = 1 

(4) λ = a

Answer: (2) λ = 1/|a|

Q3. If a = 2i + 2j − k, b = αi + βj + 2k and |a + b| = |a − b|, then 2α + 2β is equal to
(1) 2 

(2) 4 

(3) 1 

(4) 0

Answer: (1) 2

Q4. Value of λ for coplanar vectors 2i − 3j + 4k, 2i + j − k and λi − j + 2k
(1) 5 

(2) −5 

(3) 6 

(4) −6

Answer: (3) 6

Q5. The value of λ for which the vectors a = 2i + λj + k and b = i + 2j + 3k are orthogonal is
(1) 5/2 

(2) −5/2 

(3) 3/2 

(4) −3/2

Answer: (2) −5/2

Dot Product – Important Questions

These questions focus on the scalar product — the sign of a·b with respect to the angle between vectors, conditions for a sum of unit vectors to remain a unit vector, the angle when |a + b| = |a − b|, and perpendicularity conditions.

Q1. If θ is the angle between two vectors a and b, then a·b ≥ 0 only when
(1) 0 < θ < π/2 

(2) 0 ≤ θ ≤ π/2 

(3) 0 < θ < π 

(4) 0 ≤ θ ≤ π

Answer: (2) 0 ≤ θ ≤ π/2

Q2. Let a and b be two unit vectors and θ the angle between them. Then a + b is a unit vector if
(1) θ = π/4 

(2) θ = π/3 

(3) θ = π/2 

(4) θ = 2π/3

Answer: (4) θ = 2π/3

Q3. If |a + b| = |a − b|, then the angle between a and b is
(1) 0 

(2) π/4 

(3) π/2 

(4) π
Answer: (3) π/2

Q4. The vectors a = i + 3j + 5k and b = λi − j − k are perpendicular if λ is equal to
(1) −2 

(2) 2 

(3) −8 

(4) 8

Answer: (4) 8

Q5. If a = 2i + 2j − k and b = 6i − 3j + 2k, then a·b is
(1) 4 

(2) 10 

(3) 6 

(4) −6

Answer: (1) 4

Cross Product – Important Questions

These questions cover the vector product — the condition for a × b to be a unit vector, scalar triple product identities with i, j, k, the magnitude of a cross product, area of a triangle from adjacent sides, and finding a unit vector perpendicular to two given vectors.

Q1. Let 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
(1) π/6 

(2) π/4 

(3) π/3 

(4) π/2

Answer: (2) π/4

Q2. The value of i·(j × k) + j·(i × k) + k·(i × j) is
(1) 0 

(2) −1 

(3) 1 

(4) 3

Answer: (3) 1

Q3. The magnitude of the vector product of a = 2i + j − k and b = i + 2j + k is
(1) √26 

(2) √27 

(3) √30 

(4) √35

Answer: (2) √27

Q4. Area of the triangle with adjacent sides determined by a = i + 2j + k and b = 2i + j + k is

(1) √11/2 

(2) √6/2 

(3) √6 

(4) √11

Answer: (1) √11/2

Q5. The unit vector perpendicular to both vectors i + j and j + k is
(1) (i − j + k)/√3
(2) (i + j + k)/√3
(3) (i + j − k)/√3
(4) (i − j − k)/√3

Answer: (1) (i − j + k)/√3

Vector Algebra – Numerical Problems

These are trickier numerical problems, mostly taken from recent KCET papers. They involve identities linking dot and cross products, how area changes when vectors are combined linearly, and finding an unknown length or dot product value from mixed given information.

Q1. If |a × b|² + |a·b|² = 144 and |a| = 4, then |b| is equal to
(1) 3 

(2) 4 

(3) 8 

(4) 12

Answer: (1) 3

Q2. If |a| = 10, |b| = 2 and a·b = 12, then the value of |a × b| is
(1) 8 

(2) 12 

(3) 16 

(4) 4

Answer: (3) 16

Q3. |a × b|² + |a·b|² = 144 and |a| = 4, then the value of |b| is
(1) 1 

(2) 2 

(3) 3 

(4) 4

Answer: (3) 3

Q4. If |a × b| = 2√3, find the area of the parallelogram whose adjacent sides are a + b and a − b
(1) 2√3 

(2) 4√3 

(3) 6√3 

(4) 8√3

Answer: (2) 4√3

Q5. If |a| = 5, |b| = 3 and a × b = 3i + 4j, find a·b
(1) 5√2 

(2) 8 

(3) 10√2 

(4) 12
Answer: (3) 10√2

Karnataka 2nd PUC Maths Chapter 10 Vector Algebra Important Questions PDF

Students can use the Vector Algebra Important Questions PDF for additional practice. It contains formulas and 60 chapter-wise MCQs covering vector addition, magnitude, section formula, dot product and cross product, along with KCET previous year questions.

How to Use Vector Algebra Questions for Exam Preparation

Using structured solutions improves clarity and study efficiency. A consistent practice routine helps students identify weak areas early and build confidence in solving different types of vector problems. Follow these simple steps to make the best use of this study material:

• Revisit Core Concepts First – Refresh your understanding of position vectors, collinear points, and unit vectors before attempting problem sets.

• Break Down Each Solution – Work through every step of a problem rather than jumping to the final answer, as this builds a stronger grasp of the method.

• Attempt MCQs Regularly – Practising objective-type questions sharpens your speed and helps you spot the fastest route to a correct answer.

• Strengthen Numerical Practice – Dedicate time to computation-heavy problems so that lengthy calculations feel routine rather than daunting.

• Go Through Previous Year Papers – Reviewing past Vector Algebra questions helps you recognise which topics and question styles are asked most often.

• Resolve Confusion Early – Whenever a formula or step feels unclear, check a reliable solution right away instead of letting the doubt pile up.

Key Resources

• Karnataka 2nd PUC Mid Term Time Table 2026

• Karnataka 2nd PUC Mid Term Question Paper

• Karnataka 2nd PUC Previous Year Question Paper

• Karnataka 2nd PUC Exam Pattern 2027

• Karnataka 2nd PUC Syllabus 2026-27

Karnataka 2nd PUC Maths Chapter 10 covers important Vector Algebra concepts and objective questions. Regular practice of formulas, Dot Product Questions and Cross Product Questions can help students revise the chapter systematically.