2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise

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Karnataka 2nd PUC Maths Question Bank Chapter 10 Vector Algebra Miscellaneous Exercise

Question 1.
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
Answer:
Let OP be the required vector

Question 2.
Find the scalar components and magnitude of the vector joining the points P(x_1,y₁,z₁) and Q (x₂, y₂, z₂).
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Question 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.
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Question 4.
\(\overrightarrow{\mathbf{a}}=\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}, \text { then is it true that }|\overrightarrow{\mathbf{a}}|=|\overrightarrow{\mathbf{b}}|+|\overrightarrow{\mathbf{c}}|\) justify your answer.
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Question 5.
Find the value of x for which \(\mathbf{x}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \)is a unit vector.
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Question 6.
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
\(\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}} \text { and } \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\)
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Question 7.
If\(\overrightarrow{\mathrm{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}} \text { and }\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\)find a unit vector parellel to the vector
\(2 \vec{a}-\vec{b}+3 \vec{c}\)
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Question 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.
Answer:

∴ hence AC and AB are coimear. Since A s
common A, B, C arc collinear
Let B divide AC in the ratio k: I

Question 9.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a + b) and \((\vec{a}-3 \vec{b})\)externally in the ratio 1:2. Also, show that P is the mid point of the line segment RQ.
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Question 10.
The two adjacent sides of a parallelogram are \(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}} \text { and } \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}\) .Find the unit vector parallel to its diagonal. Also,find its area.
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Question 11.
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ
\(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)
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Question 12.
Let \(\overrightarrow{\mathrm{a}}=\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+7 \hat{\mathbf{k}} \text { and }\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) .find the vector \(\overrightarrow{\mathrm{d}}\) which is perpendicular to both \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}} \text { and } \overrightarrow{\mathbf{c}} . \overrightarrow{\mathbf{d}}\) = 15
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Question 13.
The scalar product of the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) a unit vector along the sum of vectors \(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}} \text { and } \quad \lambda \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is equal to ane. find the value of λ.
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Question 14.
If \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\) are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}\)
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Question 15.
Prove that
\((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \cdot(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}\) if and only if \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}\) are perpendicular ,givens \(\overrightarrow{\mathrm{a}} \neq \overrightarrow{0}, \overrightarrow{\mathrm{b}} \neq \overrightarrow{0}\)
Answer:

Choose the correct answer in Exercises 16 to 19.

Question 16.
If θ is the angle between two vectors \(\overrightarrow { { a } } \) and \(\overrightarrow { { b } } \), then 3.6 > 0 only when then \(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}} \geq \mathbf{0}\) only when
(A)\(0<\theta<\frac{\pi}{2}\)
(B)\(0 \leq \theta \leq \frac{\pi}{2}\)
(C) 0<θ<π
(D) 0≤θ≤π
Answer:

Question 17.
let \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}\) be two unit vector and θ is the angle between them. Then 3 + b is a unit vector if

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Question 18.
The value\(\hat{\mathbf{i}} \cdot(\hat{\mathbf{j}} \times \hat{\mathbf{k}})+\hat{\mathbf{j}} \cdot(\hat{\mathbf{i}} \times \hat{\mathbf{k}})+\hat{\mathbf{k}} \cdot(\hat{\mathbf{i}} \times \hat{\mathbf{j}})\)
(A) 0
(B) -1
(C) 1
(D) 3
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Question 19.
If θ is the angle between any two vectors \(\overrightarrow{\mathbf{a}} \text { and } \overrightarrow{\mathbf{b}}, \text { then }|\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}|=|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}| \)when θ is equal to
(A) 0
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{2}\)
(D) π
Answer:

2nd PUC Maths Vector Algebra Miscellaneous Exercise Extra Questions and Answers

Question 1.
Write a vector of magnitude 9 units in the direction of vector \(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) (CBSE 2019)
Answer:

Question 2.
find
\(\lambda \text { if }(2 i+6 j+4 k) \times(i-\lambda j+7 k)=0\)(CBSE 2010)
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Question 3.
If \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}}\) are two vectors such that \(|\overline{\mathbf{a}} \cdot \overline{\mathbf{b}}|=|\overline{\mathbf{a}} \times \overline{\mathbf{b}}|\), what is the angle between \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}}\)
Answer:

Question 4.
\(|\vec{a}|=\sqrt{3},|\vec{b}|=\frac{2}{3} \text { and } \vec{a} \times \vec{b}\)
is a vunit vector. Write the angle between \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}}\) (CBSE 2010)
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Question 5.
Write the projection of i – j on i + j (CBSE 2011)
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Question 6.
Write the value of \((\mathbf{i} \times \mathbf{j}) \cdot \overline{\mathbf{k}}+\mathbf{i} \cdot \mathbf{j}\)(CBSE 2012)
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Question 7.
Find the scalar components of the vector \(\overrightarrow{\mathbf{A B}}\) with initial point A (2,1) and terminal point (-5,7).     (CBSE 2012)
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Question 8.
\(\bar { a } =\vec { i } +4\vec { j } +2\vec { k } ,\quad \bar { b } =3\vec { i } -2\vec { j } +7\bar { k } ,\bar { c } =3\vec { i } -\vec { j } +4\bar { k } \) .find a vector \(\overline{\mathbf{p}}\) which is perpendicular both \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}} \text { and } \overline{\mathbf{p}} . \overline{\mathbf{c}}=18\)
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Question 9.
If \(\overline { { a } } =\overrightarrow { { i } } +\overrightarrow { { j } } +\overline { { k } } ,\quad \overline { { b } } =4\overrightarrow { { i } } -2\overrightarrow { { j } } +3\overline { { k } } ,\overline { { c } } =\overrightarrow { { i } } -2\overrightarrow { { j } } +\overline { { k } } \),find a vector of magnitude 6 units which is parallel to the vector \(2\overline { { a } } -\overline { b } +\overline { { 3c } } \)
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Question 10.
\(\overline { { a } } =\overrightarrow { { i } } +4\overrightarrow { { j } } +2\overline { { k } } ,\quad \overline { { b } } =3\overrightarrow { { i } } -2\overrightarrow { { j } } +7\overline { { k } } ,\overline { { c } } =2\overrightarrow { { i } } -\overrightarrow { { j } } +4\overline { { k } } \). find a vector d such that it is perpendiular to both \(\overline{\mathbf{a}} \text { and } \overline{\mathbf{b}} \text { and }\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{d}}=18\)
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Question 11.
Using vectors find the area of the triangle with vectors A (1, 1, 2) , B (2, 3, 5), C(1, 5, 5) .   (CBSE 2011)
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Question 12.
\(\alpha=3 i – j, \bar{\beta}=2 i+j-3 \bar{k}\) Express β in the form the form
β = β₁ + β₂ where β₁|| α and β₂ ⊥α (CBSE 2012)
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Question 13.
\(\begin{aligned}&\overline{\mathbf{a}} \times \overline{\mathbf{b}}=\overline{\mathbf{c}} \times \overline{\mathbf{d}}, \quad \overline{\mathbf{a}} \times \overline{\mathbf{c}}=\overline{\mathbf{b}} \times \overline{\mathbf{d}} \text { Show that }\\&(\overline{\mathbf{a}}-\overline{\mathbf{d}}) \| \mathbf{t} \mathbf{o}-\overline{\mathbf{c}} \text { when } \overline{\mathbf{a}} \neq \overline{\mathbf{d}} \text { and } \overline{\mathbf{b}} \neq \overline{\mathbf{c}}\end{aligned}\)(ISE 2011)
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Question 14.
find
\(\overline{\mathbf{a}} \cdot \overline{\mathbf{b}} \text { of }|\overline{\mathbf{a}}|=\mathbf{3},|\overline{\mathbf{b}}|=\mathbf{4},|\overline{\mathbf{a}} \times \mathbf{b}|=6\) (CBSE 2009)
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Question 15.
If \(\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}\) represent the sides of Δ show that \(\overline{\mathbf{a}} \times \overline{\mathbf{b}}=\overline{\mathbf{b}} \times \overline{\mathbf{c}}=\overline{\mathbf{c}} \times \overline{\mathbf{a}}\) hence derive sine formula. (CBSE 2008)
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