<p><strong>Let <math><mrow><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>=</mo><mi>α</mi><mover accent="true"><mi>i</mi><mo>^</mo></mover><mi>a</mi><mo>+</mo><mn>3</mn><mover accent="true"><mi>j</mi><mo>^</mo></mover><mo>−</mo><mover accent="true"><mi>k</mi><mo>^</mo></mover><mo>,</mo><mtext> </mtext><mtext> </mtext><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>=</mo><mn>3</mn><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>−</mo><mi>β</mi><mover accent="true"><mi>j</mi><mo>^</mo></mover><mo>+</mo><mn>4</mn><mover accent="true"><mi>k</mi><mo>^</mo></mover><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mover accent="true"><mi>c</mi><mo>→</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><mtext> </mtext><mtext> </mtext><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mtext> </mtext><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>R</mi><mo>,</mo></mrow></math> be three vectors. If the projection of <math><mrow><mover accent="true"><mi>a</mi><mo>→</mo></mover><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>n</mi><mtext> </mtext><mtext> </mtext><mover accent="true"><mi>c</mi><mo>→</mo></mover><mtext> </mtext><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>1</mn><mn>0</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>=</mo><mo>−</mo><mn>6</mn><mover accent="true"><mi>i</mi><mo>^</mo></mover><mo>+</mo><mn>1</mn><mn>0</mn><mover accent="true"><mi>j</mi><mo>^</mo></mover><mo>+</mo><mn>7</mn><mover accent="true"><mi>k</mi><mo>^</mo></mover><mo>,</mo></mrow></math> then the value of + is equal to:</strong></p>

a→⋅c^=α+6+21+4+4103=8+α3⇒α=2b→×c→=i^ (2β−8)+j^ (10)+k^ (6+β)=−6i^+10j^+7k^So = 1

Ask & Answer Question

Let $\vec{a} = α \hat{i} a + 3 \hat{j} - \hat{k}, \vec{b} = 3 \hat{i} - β \hat{j} + 4 \hat{k} a n d \vec{c} = \hat{i} + 2 \hat{j} - 2 \hat{k} w h e r e α, β \in R,$ be three vectors. If the projection of $\vec{a} o n \vec{c} i s \frac{10}{3} a n d \vec{b} \times \vec{c} = - 6 \hat{i} + 10 \hat{j} + 7 \hat{k},$ then the value of + is equal to:

Option 1 -

Option 2 -

Option 3 -

Option 4 -

3 Views | Posted 2 months ago

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

$\vec{a} \cdot \hat{c} = \frac{α + 6 + 2}{\sqrt[]{1} + 4 + 4}$

$\frac{10}{3} = \frac{8 + α}{3} \Rightarrow α = 2$

$\vec{b} \times \vec{c} = \hat{i} (2 β - 8) + \hat{j} (10) + \hat{k} (6 + β)$

$= - 6 \hat{i} + 10 \hat{j} + 7 \hat{k}$

So = 1

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