# Exam Question for Class 12 Mathematics Chapter 10 Vector Algebra

Please refer to below Exam Question for Class 12 Mathematics Chapter 10 Vector Algebra. These questions and answers have been prepared by expert Class 12 Mathematics teachers based on the latest NCERT Book for Class 12 Mathematics and examination guidelines issued by CBSE, NCERT, and KVS. We have provided Class 12 Mathematics exam questions for all chapters in your textbooks. You will be able to easily learn problems and solutions which are expected to come in the upcoming class tests and exams for standard 10th.

## Chapter 10 Vector Algebra Class 12 Mathematics Exam Question

All questions and answers provided below for Exam Question Class 12 Mathematics Chapter 10 Vector Algebra are very important and should be revised daily.

**Exam Question Class 12 Mathematics Chapter 10 Vector Algebra**

**Very ****Short Answer Type Questions**

**Question**. Find a vector a̅**of magnitude 5 √2, making an** **angle of** **π**/**4** **with x-axis, π**/**2** **with y-axis and an** **acute angle θ with z-axis.****Answer.**

**Question**. Find the unit vector in the direction of the sum**of the vectors 2iˆ + 3jˆ − kˆ and 4iˆ − 3jˆ + 2k****Answer.**

**Question**. Find a vector in the direction of a̅ = î – 2ĵ that**has magnitude 7 units.****Answer.** A unit vector in the direction of a̅ = î – 2

**Question**. Write the direction ratios of the vector 3a̅ + 2b̅ where**Answer.**

**Question**. Find the value of p for which the vectors**3iˆ + 2 jˆ + 9kˆ and iˆ − 2pjˆ + 3kˆ are parallel.****Answer.**

**Question**. Find a vector in the direction of vector 2iˆ − 3jˆ + 6kˆ which has magnitude 21 units.**Answer.**

**Question**. Write a unit vector in the direction of vector PQ, where P and Q are the points(1, 3, 0) and**(4, 5, 6) respectively.****Answer.**

**Question.**

**Answer.** Given, a̅ = b̅

**Question**. Find the scalar components of the vector AB^{— }with initial point A(2, 1) and terminal point**B(–5, 7).****Answer.**

**Question**. Find a unit vector parallel to the sum of the**vectors iˆ + jˆ + kˆ and 2iˆ − 3jˆ + 5k****Answer.**

**Question**. Find a unit vector in the direction of the vector**a̅ = 3iˆ − 2 jˆ + 6kˆ .****Answer.** The given vector is a̅ = 3iˆ − 2 jˆ + 6k

**Question**. Write the direction cosines of the vector**−2iˆ + jˆ − 5kˆ. ****Answer.** We have, a̅ = −2iˆ + jˆ − 5kˆ

Direction cosines of the given vector are

**Question**. For what value of ‘a’, the vectors 2iˆ − 3jˆ + 4kˆ**and aiˆ + 6 jˆ − 8kˆ are collinear. ****Answer.** We have, 2iˆ − 3jˆ + 4kˆ and aiˆ + 6 jˆ − 8kˆ

Two vectors are collinear if and only if,

**Question**. Write a unit vector in the direction of the vector**a̅ = 2iˆ + jˆ + 2kˆ****Answer.** a̅ = 2iˆ + jˆ + 2kˆ

**Question**. Write a vector of magnitude 15 units in the**direction of vector iˆ − 2 jˆ + 2kˆ.****Answer.** We have, a̅ = iˆ − 2 jˆ + 2kˆ

The unit vector in the direction of given vector a̅ is

**Question**. Find the position vector of the mid-point of the line-segment AB, where A is the point (3, 4, –2)**and B is the point (1, 2, 4).****Answer.**

**Question**. Write the value of p for which a̅ = 3iˆ + 2 jˆ + 9kˆ**and b̅ = iˆ + pjˆ + 3kˆ are parallel vectors.****Answer.** We have, a̅ = 3iˆ + 2 jˆ + 9kˆ and b̅ = iˆ + pjˆ + 3kˆ Two vectors are parallel if

**Question**. Write a unit vector in the direction of**a̅ = 2iˆ − 6 jˆ + 3k****Answer.** We have,

**Question**. If P(1, 5, 4) and Q(4, 1, –2), then find the**direction ratios of PQ**^{—}**Answer.** We have, P(1, 5, 4) and Q(4, 1, – 2) Since the vector is to be directed from P to Q, clearly P is the initial point and Q is the terminal point.

**Question**. If a̅ = iˆ + 2 jˆ − kˆ and**b̅ = 3iˆ + jˆ − 5kˆ , then find a** **unit vector in the direction of a̅, – b̅,****Answer.**

**Question**. Write the number of vectors of unit**length perpendicular to both the vectors** **a̅ = 2î + ĵ + 2k̂ and b̅ = ĵ + k̂.**

Answer. Given, a̅ = 2î + ĵ + 2k̂ and b̅ = ĵ + k̂.

Unit vectors perpendicular to

**So, there are two unit vectors perpendicular to the given vectors.**

**Question**.

**Answer.**

**Question.**

**Answer.**

**Question**. Find the angle between x-axis and the vector**iˆ + jˆ + kˆ.****Answer.** Here, a̅ =iˆ + jˆ + kˆ and vector along x-axis is iˆ .

∴ Angle between a̅ and iˆ is given by

**Question**. Find |x̅| if for a unit vector a̅ ,

**Answer.**

**Question**. Write the value of λ so that the vectors**a̅ =2iˆ +λjˆ +kˆ and** **b̅ = iˆ −2 jˆ +3kˆ are** **perpendicular to each other.****Answer.**

**Question**. Write the projection of**b̅ + c̅ on a̅, where** **a̅ = 2iˆ −2 jˆ + kˆ, b̅ = iˆ +2 jˆ −2kˆ and c̅ = 2iˆ − jˆ + 4k̂****Answer.** a̅ = 2iˆ −2 jˆ + kˆ, b̅ = iˆ +2 jˆ −2kˆ and c̅ = 2iˆ − jˆ + 4k̂

**Question**. Find ‘λ’ when the projection of

**Answer.**

**Question**. Write the value of (î x ĵ) · kˆ + î · ĵ. **Answer.** (î x ĵ) · kˆ + î · ĵ = kˆ · kˆ + iˆ · jˆ = 1 + 0 = 1

**Question**. Write the value of (kˆ x ĵ) · î + ĵ · kˆ**Answer.** (kˆ x ĵ) · î + ĵ ·kˆ = −iˆ · iˆ + jˆ · kˆ = −1 + 0 = −1

**Question**. Write the value of (kˆ x î) · ĵ + î · kˆ**Answer.** ** **(kˆ x î) · ĵ + î · kˆ = ĵ · ĵ +0 = 1 + 0 = 1

**Question**. Write the angle between two vector a̅ and b̅ with magnitudes √3 and 2 respectively having**a̅ · b̅ = √6.****Answer.** Let θ be the angle between a̅ and b̅, then

**Question**. Write the projection of the vector iˆ − jˆ on the**vector iˆ + jˆ.****Answer.**

**Question.**

**Answer.**

**Question**. If a̅ and b̅ are two vectors such that |a̅ · b̅| = |a̅ x b̅| then find the angle between a̅ and b̅**Answer.**

**Question**. Find λ, if (2î + 6 ĵ +14k̂) λ (î − ĵ + 7k̂) = 0.**Answer.**

**Question**. If |a̅| = 2, |b̅| = 3 and a̅ · b̅ = 2 then find the projection of b̅ on a̅**Answer.**

**Question**. Find the angle between two vectors a̅ and b̅**with magnitudes 1 and 2 respectively and when |a̅ x b̅| = √3**** Answer.** We have,

**Question**. Find the projection of a̅ on b̅, if a̅ · b̅ = 8 and**b̅ = 2iˆ + 6 jˆ + 3kˆ.****Answer.**

**Question**. Find the value of p, if**(2î+6ĵ+27k̂)x(î+3ĵ+ pk̂) = 0****Answer.**

**Question**.

**Answer.**

**Question**. Find a̅ ·b̅, if a̅ =−iˆ + jˆ −2kˆ and b̅ = 2iˆ +3jˆ −kˆ.**Answer.**

**Question**. Find the angle between the vectors a̅ = iˆ − jˆ + kˆ and b̅ = iˆ + jˆ − kˆ.

Answer.

**Question**.

**Answer.**

**Question.**

**Answer.** We have,

**Question**. Find the value of the following :**î · ( ĵ x k̂ ) + ĵ · (î x k̂ ) + k̂ · (î x ĵ)****Answer.**

**Question**, Find the projection of**b̅ + c̅ on a̅ where** **a̅ = iˆ + 2 jˆ + kˆ ,b̅ = iˆ + 3jˆ + kˆ and c̅ = iˆ + kˆ**

**Answer.**

**Question.**

**Answer.** Since the vectors are coplanar.

⇒ 1(–3 + λ)–3 (6 – 0) + 1(2λ – 0) = 0

⇒ –3 + λ – 18 + 2λ = 0

⇒ 3λ – 21 = 0 ⇒ λ = 7

**Question. **

**Answer.**

** Question. Find the value of λ which makes the**

**Answer.** Since vectors, a̅, b̅, c̅, are coplanar

⇒ 2(10 – 3λ) + 1(5+9) + 1(–λ – 6) = 0

⇒ 20 – 6λ + 14 – λ – 6 = 0

⇒ –7λ + 28 = 0 ⇒ λ = 4

**Short Answer Type Questions**

**Question.**

**Answer.**

**Question**. If a̅ b̅ c̅, are three mutually perpendicular vectors of the same magnitude, prove that**a̅ + b̅ + c̅, is equally inclined with the vectors** **a̅, b̅ and c̅** **Answer.**

**Question**. Dot product of a vector with vectors iˆ − jˆ + kˆ, 2iˆ + jˆ − 3kˆ and iˆ + jˆ + kˆ are respectively 4, 0 and 2. Find the vector.**Answer.** Let the required vector be

**Question**. Find a vector of magnitude 5 units and**perpendicular to each of the vectors (a̅ + b̅) and (a̅ – b̅) where a̅ = iˆ + jˆ + kˆ and** **b̅ = iˆ + 2 jˆ + 3kˆ**** Answer.** a̅ = iˆ + jˆ + kˆ and b̅ = iˆ + 2 jˆ + 3kˆ

**Question**. Show that the vectors a̅, b̅ and c̅ are coplanar if**a̅ + b̅, b̅ + c̅ and c̅ + a̅ are coplanar.**** Answer.** Since, a̅ + b̅, b̅ + c̅ and c̅ + a̅ are coplanar

**Question**. Find the value of λ so that the four points A, B, C and D with position vectors 4iˆ +5jˆ +kˆ, − jˆ −kˆ , 3iˆ +λjˆ +4k and −4iˆ +4 jˆ +4kˆ**respectively are coplanar.**** Answer.** Here position vectors of A, B, C and D

**Question**. Find the position vector of a point R which**divides the line joining two points P and** **Q whose position vectors are (2a̅ + b̅) and a̅ + b̅** **a̅ − 3b̅ respectively, externally in the ratio** **1 : 2. Also, show that P is mid-point of the line** **segment RQ.****Answer.** The position vector of R is given by

**Question.**

**Answer.**

**Question**. The two adjacent sides of a parallelogram are 2î − 4ĵ − 5k̂ and 2î + 2ĵ + 3k̂ . Find the two**unit vectors parallel to its diagonals. Using the diagonal vectors, Find the area of the** **parallelogram.****Answer.** Let a̅ = 2î − 4ĵ − 5k̂ and 2î + 2ĵ + 3k̂

**Question**.

**Answer.** Two non zero vectors are parallel if and only if their cross product is zero vector.

So, we have to prove that cross product of a̅ − d̅ and b̅ − c̅ is zero vector.

**Question**. Prove that : [a̅,b̅ + c,d̅][a̅,b̅,d̅] + [a̅,c,d̅].**Answer.**

**Question**. Prove that, for any three vectors **a̅** **b̅** c̅,

**Answer.**

**Question**. Find the values of λ for which the angle**between the vectors a̅ = 2λ ^{2} iˆ + 4λjˆ + kˆ and**

**b̅ = 7iˆ − 2 jˆ + λkˆ is obtuse.**

**Answer.**

**Question**. If a̅ ,b̅ and c̅ are three vectors such that each one is perpendicular to the vector**obtained by sum of the other two and**

**Answer.**

**Question.**

**Answer.**