CUET PG 2022 — Mathematics PYQ

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Question

CUET PG 2022 — Mathematics PYQ

CUET PG | Mathematics | 2022

If $a = \hat{i} + \hat{j} + \hat{k}$ , $a \cdot b = 1$ and $a \times b = \hat{j} \hat{k}$ , then $b$ is equal to:

Choose the correct answer:

Explanation

1. Use the Vector Triple Product Formula:

We know the identity:

\vec{a} \times (\vec{a} \times \vec{b}) = (\vec{a} \cdot \vec{b})\vec{a} - (\vec{a} \cdot \vec{a})\vec{b}

2. Calculate the components needed:

First, find $\vec{a} \cdot \vec{a}$ (which is $|\vec{a}|^2$ ):

\vec{a} \cdot \vec{a} = (1)^2 + (1)^2 + (1)^2 = 3

Next, calculate the Left Hand Side (LHS) $\vec{a} \times (\vec{a} \times \vec{b})$ :

\vec{a} \times (\hat{j} - \hat{k}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 0 & 1 & -1 \end{vmatrix}

= \hat{i}(-1 - 1) - \hat{j}(-1 - 0) + \hat{k}(1 - 0)

= -2\hat{i} + \hat{j} + \hat{k}

3. Substitute values into the identity:

Now, plug everything back into $(\vec{a} \cdot \vec{b})\vec{a} - (\vec{a} \cdot \vec{a})\vec{b} = \vec{a} \times (\vec{a} \times \vec{b})$ :

(1)(\hat{i} + \hat{j} + \hat{k}) - 3\vec{b} = -2\hat{i} + \hat{j} + \hat{k}

4. Solve for $\vec{b}$ :

-3\vec{b} = (-2\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + \hat{j} + \hat{k})

-3\vec{b} = -3\hat{i} + 0\hat{j} + 0\hat{k}

-3\vec{b} = -3\hat{i}

Divide by $-3$ :

\vec{b} = \hat{i}

Choose the correct answer:

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