2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3

Students can Download Maths Chapter 10 Vector Algebra Ex 10.3 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3

2nd PUC Maths Vector Algebra NCERT Text Book Questions and Answers Ex 10.3

Question 1.
Find the angle between two vectors \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}\) with magnitudes \(\sqrt{3}\) and 2 respectively having \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\sqrt{6}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.1

KSEEB Solutions

Question 2.
Find the angle between the vectors
\(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \text { and } 3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.2

Question 3.
Find the projection of the vector \(\hat{\mathbf{i}}-\hat{\mathbf{j}}\) on the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.3

Question 4.
Find the projection of the vector \(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}\) on the vector \(7 \hat{\mathbf{i}}-\hat{\mathbf{j}}+8 \hat{\mathbf{k}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.4

Question 5.
Show that each of the given three vectors is a unit vector.
\(\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \frac{1}{7}(3 \hat{i}-6 \hat{j}+2 \hat{k}), \frac{1}{7}(6 \hat{i}+2 \hat{j}-3 \hat{k})\)
Also, show that they are mutually perpendicular to each other.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.5

KSEEB Solutions

Question 6.
\(\text { Find }|\overrightarrow{\mathrm{a}}| \text { and }|\overrightarrow{\mathrm{b}}|, \text { if }(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}) \cdot(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})=8 \text { and }|\overrightarrow{\mathrm{a}}|=8|\overrightarrow{\mathrm{b}}|\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.6

Question 7.
Evaluate the product
\((3 \vec{a}-5 \vec{b}) \cdot(2 \vec{a}+7 \vec{b})\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.7

Question 8.
Find the magnitude of two vectors \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}\), having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.8

Question 9.
Find \(|\overrightarrow{\mathbf{x}}|\) if for a unit vector \(\overrightarrow{\mathrm{a}},(\overrightarrow{\mathrm{x}}-\overrightarrow{\mathrm{a}})\cdot(\overrightarrow{\mathrm{x}}+\overrightarrow{\mathrm{a}})=12\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.9
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.10

Question 10.
If \(\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{c}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}\) are such that \(\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}}\) is perpendicular to \(\overrightarrow{\mathbf{c}}\), then the value of λ
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.11

KSEEB Solutions

Question 11.
Show that \(|\overrightarrow{\mathbf{a}}| \overrightarrow{\mathbf{b}}+|\overrightarrow{\mathbf{b}}| \overrightarrow{\mathbf{a}}\) is a perpendicular to \(|\overrightarrow{\mathrm{a}}| \overrightarrow{\mathbf{b}}-|\overrightarrow{\mathrm{b}}| \overrightarrow{\mathrm{a}}\), for ant two non zero vectors \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.11

Question 12.
If \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}=0 \text { and } \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=0\), then what can be concluded about the vector \(\overrightarrow{\mathbf{b}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.13

Question 13.
If \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\)are unit vector such that \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), find the value of \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.14
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.15
Question 14.
If either vector \(\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{0}} \quad \text { or } \quad \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{0}}\) then \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{0}}\). But the converse need not be true. Justify your answer with an example.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.16

KSEEB Solutions

Question 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 \(\bar { BA }\) and \(\bar { BC }\) .
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.17

Question 16.
Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, -1) are collinear.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.18
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.19

Question 17.
Show that the vector\(2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}\) and
\(\text { and } 3 \hat{i}-4 \hat{j}-4 \hat{k}\) from the vertices of a right angled triangle:
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.20

KSEEB Solutions

Question 18.
If \(\overrightarrow{\mathbf{a}}\) is a nonzero vector of magnitude ‘a’ and
λ a nonzero scalar then λ \(\vec{a}\) is unit vector if……….
(A) λ = 1
(B) λ = -1
(C) a = |λ|
(D) a = 1/|λ|
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.3.21

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