Q

# Help me answer: - Vector Algebra - JEE Main

If $\vec{a},\; \vec{b},\: and\: \vec{c}$ are unit vectors such that $\vec{a}+2 \vec{b}+2\vec{c}=\vec{0},\; \; then\: \: \left | \vec{a }\times \vec{c} \right |$ is equal to :

• Option 1)

$\frac{\sqrt{15}}{4}$

• Option 2)

$\frac{1}{4}$

• Option 3)

$\frac{15}{16}$

• Option 4)

$\frac{\sqrt{15}}{16}$

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As we learned

Scalar Product of two vectors (dot product) -

$\vec{a}\vec{b}=\left | a \right |\left | b \right |Cos\theta$

- wherein

$\Theta$ is the angle between the vectors$\vec{a}\: and\:\vec{b}$

Vector Product of two vectors(cross product) -

If $\vec{a}$ and $\vec{b}$ are two vectors and $\Theta$ is the angle between them , then $\vec{a}\times \vec{b}=\left |\vec{a} \left | \right |\vec{b} \right |Sin\Theta \hat{n}$

- wherein

$\hat{n}$ is unit vector perpendicular to both $\vec{a} \: and \: \vec{b}$

Given $\vec{a},\vec{b},\vec{c}$ are unit vectors.

$2\vec{b} =-\left ( \vec{a}+2\vec{c} \right )$

$4=1+4+4\vec{a}\vec{c}\Rightarrow \cos \theta =-\frac{1}{4}$

$\tan \left | \vec{a} \times \vec{c}\right |=\left | \vec{a} \right |\left | \vec{c} \right |\sin \theta$

$=1\times 1\times \frac{\sqrt{15}}{4}$

Option 1)

$\frac{\sqrt{15}}{4}$

Option 2)

$\frac{1}{4}$

Option 3)

$\frac{15}{16}$

Option 4)

$\frac{\sqrt{15}}{16}$

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