A vector <math><mrow><mover accent="true"><mrow><mi>a</mi></mrow><mo stretchy="true">→</mo></mover></mrow></math> is parallel to the line of intersection of the plane determined by the vectors <math><mrow><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>,</mo><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>j</mi><mo>^</mo></mover></mrow></math> and the plane determined by the vectors <math><mrow><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>−</mo><mover accent="true"><mi>j</mi><mo>^</mo></mover><mo>,</mo><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>k</mi><mo>^</mo></mover><mo>.</mo></mrow></math> The obtuse angle between <math><mrow><mover accent="true"><mrow><mi>a</mi></mrow><mo stretchy="true">→</mo></mover></mrow></math> and the vector <math><mrow><mover accent="true"><mrow><mi>b</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>−</mo><mn>2</mn><mover accent="true"><mi>j</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mover accent="true"><mi>k</mi><mo>^</mo></mover></mrow></math> is

If n^1 is a vector normal to the plane determined by i^ and i^+j^ then n^1=|i^j^k^100110|=k^If n^2 is a vector normal to the pane determined by i^−j^, and i^+k^ then n^2 = |i^j^k^1−10101|=i^−j^+k^Vector a^ is parallel to n^1×n^2 i.e. a^ is parallel to |i^j^k^001−1−11|=i^−j^Given b→=i^−2j^+2k^consine of acute angle between a^ and b^ = |a^.b^|a^|.|b^||=12obtuse angle between a^ and b^=3π4

A vector&nbsp;<math><mrow><mover accent="true"><mrow><mi>a</mi></mrow><mo stretchy="true">→</mo></mover></mrow></math>&nbsp;is parallel to the line of intersection of the plane determined by the vectors&nbsp;<math><mrow><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>,</mo><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>j</mi><mo>^</mo></mover></mrow></math>&nbsp;and the plane determined by the vectors&nbsp;<math><mrow><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>−</mo><mover accent="true"><mi>j</mi><mo>^</mo></mover><mo>,</mo><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>k</mi><mo>^</mo></mover><mo>.</mo></mrow></math>&nbsp;The obtuse angle between&nbsp;<math><mrow><mover accent="true"><mrow><mi>a</mi></mrow><mo stretchy="true">→</mo></mover></mrow></math>&nbsp;and the vector&nbsp;<math><mrow><mover accent="true"><mrow><mi>b</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>−</mo><mn>2</mn><mover accent="true"><mi>j</mi><mo>^</mo></mover><mo>+</mo><mn>2</mn><mover accent="true"><mi>k</mi><mo>^</mo></mover></mrow></math>&nbsp;is

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Maths Class 12th Vector Algebra Maths NCERT Exemplar Solutions Class 12th Chapter Ten

A vector $\vec{a}$ is parallel to the line of intersection of the plane determined by the vectors $\hat{i}, \hat{i} + \hat{j}$ and the plane determined by the vectors $\hat{i} - \hat{j}, \hat{i} + \hat{k} .$ The obtuse angle between $\vec{a}$ and the vector $\vec{b} = \hat{i} - 2 \hat{j} + 2 \hat{k}$ is

Option 1 -

$\frac{3 π}{4}$

Option 2 -

$\frac{2 π}{3}$

Option 3 -

$\frac{4 π}{5}$

Option 4 -

$\frac{5 π}{6}$

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

$I f {\hat{n}}_{1}$ is a vector normal to the plane determined by $\hat{i} a n d \hat{i} + \hat{j} t h e n {\hat{n}}_{1} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix} | = \hat{k}$

$I f {\hat{n}}_{2}$ is a vector normal to the pane determined by $\hat{i} - \hat{j},$ and $\hat{i} + \hat{k}$ then ${\hat{n}}_{2}$ = $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 1 & 0 \\ 1 & 0 & 1 \end{matrix} | = \hat{i} - \hat{j} + \hat{k}$

Vector $\hat{a}$ is parallel to ${\hat{n}}_{1} \times {\hat{n}}_{2}$ i.e. $\hat{a}$ is parallel to $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ - 1 & - 1 & 1 \end{matrix} | = \hat{i} - \hat{j}$

Given $\vec{b} = \hat{i} - 2 \hat{j} + 2 \hat{k}$

consine of acute angle between $\hat{a} a n d \hat{b}$ = $| \frac{\hat{a} . \hat{b}}{| \hat{a} | . | \hat{b} |} | = \frac{1}{\sqrt[]{2}}$

obtuse angle between $\hat{a} a n d \hat{b} = \frac{3 π}{4}$

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A vector a→ is parallel to the line of intersection of the plane determined by the vectors i^,i^+j^ and the plane determined by the vectors i^−j^,i^+k^. The obtuse angle between a→ and the vector b→=i^−2j^+2k^ is

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