2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4

Students can Download Maths Chapter 10 Vector Algebra Ex 10.4 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 10.4

2nd PUC Maths Vector Algebra NCERT Text Book Questions and Answers Ex 10.4

Question 1.
Find
\(|\vec{a} \times \vec{b}|, \text { if } \vec{a}=\hat{i}-7 \hat{j}+7 \hat{k} \text { and }\overrightarrow{\mathbf{b}}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.1
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.2

Question 2.
Find a unit vector perpendicular to each of the vector \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} \quad \text { and } \quad \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}\) where \(\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}, \text { and } \vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.3

KSEEB Solutions

Question 3.
If a unit vector \(\overrightarrow{\mathrm{a}}\) makes angles \(\frac{\pi}{3} \text { with } \hat{\mathbf{i}}, \frac{\pi}{4}\) with \(\hat{\mathbf{j}}\) and acute angle θ with \(\hat{\mathbf{k}}\) then find θ and hence, the component of \(\overrightarrow{\mathrm{a}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.4
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.5

Question 4.
Show that
\((\overline{\mathbf{a}}-\overline{\mathbf{b}}) \times(\overline{\mathbf{a}}+\overline{\mathbf{b}})=\mathbf{2}(\overline{\mathbf{a}} \times \overline{\mathbf{b}})\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.6

Question 5.
Find λ and μ if
\((2 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}+27 \hat{\mathbf{k}}) \times(\hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\mu \hat{\mathbf{k}})=\overrightarrow{\mathbf{0}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.7

KSEEB Solutions

Question 6.
Given that \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\mathbf{0} \text { and } \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{0}}\). when can you conclude about the vectors \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.8
either \(\bar{a}=0 \text { or } \bar{b}=0\)
They can’t be parallel and perpendicular at the same time.
∴ At least one of \(\bar{a} \text { and } \bar{b} \)is zero  vector.

Question 7.
Let the vector \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) be given as \(\mathbf{a}_{1} \hat{\mathbf{i}}+\mathbf{a}_{2} \hat{\mathbf{j}}+\mathbf{a}_{3} \hat{\mathbf{k}}, \mathbf{b}_{1} \hat{\mathbf{i}}+\mathbf{b}_{2} \hat{\mathbf{j}}+\mathbf{b}_{3} \hat{\mathbf{k}}, \mathbf{c} \hat{\mathbf{i}}+\mathbf{c}_{2} \hat{\mathbf{j}}+\mathbf{c}_{3} \hat{\mathbf{k}}\). Then show that \(\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})=\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.9

Question 8.
If either\(\overrightarrow{\mathrm{a}}=\overrightarrow{0} \text { or } \overrightarrow{\mathrm{b}}=\overrightarrow{0}, \text { then } \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{0}\). Is the converse true? Justify your answer with an example.
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.10

KSEEB Solutions

Question 9.
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.11

2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.12

Question 10.
Find the area of the parallelogram whose adjacent sides are determined by the vectors a
\(\overrightarrow{\mathrm{a}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}} \text { and } \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+\hat{\mathbf{k}}\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.13

Question 11.
Let the vector \(\overrightarrow{\mathbf{a}} \text { and }\overrightarrow{\mathbf{b}}\) be such that \(|\vec{a}|=3 \text { and }|\vec{b}|=\frac{\sqrt{2}}{3}, \text { then } \vec{a} \times \vec{b}\) is a unit vector ,if angle between \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}} \text { is }\)
(A) \(\pi / 6\)
(B) \(\pi / 4\)
(C) \(\pi / 3\)
(D) \(\pi / 2\)
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.14

KSEEB Solutions

Question 12.
Area of a rectangle having vertices A, B, C and D with position vectors
\(\begin{aligned}&-\hat{\mathbf{i}}+\frac{1}{2} \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \hat{\mathbf{i}}+\frac{1}{2} \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \hat{\mathbf{i}}-\frac{1}{2} \hat{\mathbf{j}}+4 \hat{\mathbf{k}} \text { and }\\&-\hat{\mathbf{i}}-\frac{1}{2} \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \text { respectively is }\end{aligned}\)
(A) \(\frac{1}{2}\)
(B) 1
(C) 2
(D) 3
Answer:
2nd PUC Maths Question Bank Chapter 10 Vector Algebra Ex 10.4.15

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