If {a}={i}+{j}+{k}, {b}={j}-{k} then a vector {c} such that {a} × {c}={b} and {a} · {c}=3 is

Correct Answer is : 53i^+23j^+23k^5/3 {{i}}+2/3 {{j}}+2/3 {{k}}35i^+32j^+32k^

Correct Answer is : 53i^+23j^+23k^5/3 {{i}}+2/3 {{j}}+2/3 {{k}}35i^+32j^+32k^

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MHTCET 25 Apr 2025 Shift 1 Paper

Question : 105 of 150

Marks: +1, -0

\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{j}-\hat{k}

\bar{c}

\overline{a} \times \overline{c}=\overline{b}

\overline{a} \cdot \overline{c}=3

$\frac{5}{3} \hat{\mathrm{i}}+\frac{2}{3} \hat{\mathrm{j}}+\frac{2}{3} \hat{\mathrm{k}}$
$\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$
$\hat{\mathrm{i}}+2 \hat{\mathrm{k}}$
$\quad 2 \hat{i}-3 \hat{j}+4 \hat{k}$

Solution:

Concept:

Use

\bar{a} \times \bar{c} = \bar{b}

and

\bar{a} \cdot \bar{c} = 3

to form equations for the unknown components of

\bar{c}

Explanation:

Let

\bar{c} = p\hat{i} + q\hat{j} + r\hat{k}

Given

\bar{a} = \hat{i} + \hat{j} + \hat{k}

and

\bar{b} = \hat{j} - \hat{k}

Compute the cross product:

\bar{a} \times \bar{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ p & q & r \end{vmatrix}

= \hat{i}(r-q) - \hat{j}(r-p) + \hat{k}(q-p)

= \hat{i}(r-q) + \hat{j}(p-r) + \hat{k}(q-p)

Since

\bar{a} \times \bar{c} = \hat{j} - \hat{k}

, equate components:

r - q = 0 \quad \Rightarrow \quad r = q

p - r = 1

q - p = -1 \quad \Rightarrow \quad p - q = 1

Now use the dot product:

\bar{a} \cdot \bar{c} = (\hat{i}+\hat{j}+\hat{k})\cdot(p\hat{i}+q\hat{j}+r\hat{k}) = 3

\Rightarrow p + q + r = 3

Substitute

r = q

into

p - q = 1

, so

p = q + 1

Then

p + q + r = (q+1) + q + q = 3

gives

3q + 1 = 3

, hence

q = \frac{2}{3}

r = \frac{2}{3}

and

p = \frac{5}{3}

Answer:

\bar{c} = \frac{5}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}

Option A is correct.

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