SATHEE: Vectors 3 Question 7

Vectors 3 Question 7

7. If $\vec{V} = 2 \vec{i} + \vec{j} - \vec{k}$ and $\vec{W} = \vec{i} + 3 \vec{k}$ . If $\vec{U}$ is a unit vector, then the maximum value of the scalar triple product $[\vec{U} \vec{V} \vec{W}]$ is

(2002, 1M)

(a) -1

(b) $\sqrt{10} + \sqrt{6}$

(d) $\sqrt{60}$

Show Answer

Answer:

Correct Answer: 7. (c)

Solution:

Given, $\vec{V} = 2 \hat{i} + \hat{j} - \hat{k}$ and $\vec{W} = \hat{i} + 3 \hat{k}$

$\begin{aligned} [\vec{U} \vec{V} \vec{W}] & = \vec{U} \cdot [(2 \hat{i} + \hat{j} - \hat{k}) \times (\hat{i} + 3 \hat{k})] \\ = \vec{U} \cdot (3 \hat{i} - 7 \hat{j} - \hat{k}) = | \vec{U} | | 3 \hat{i} - 7 \hat{j} - \hat{k} | \cos θ \end{aligned}$

which is maximum, if angle between $\vec{U}$ and $3 \hat{i} - 7 \hat{j} - \hat{k}$ is 0 and maximum value

$= | 3 \hat{i} - 7 \hat{j} - \hat{k} | = \sqrt{59}$

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Vectors 3 Question 7

7. If V→=2i→+j→−k→ and W→=i→+3k→. If U→ is a unit vector, then the maximum value of the scalar triple product [U→V→W→] is

Answer:

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7. If $\vec{V} = 2 \vec{i} + \vec{j} - \vec{k}$ and $\vec{W} = \vec{i} + 3 \vec{k}$ . If $\vec{U}$ is a unit vector, then the maximum value of the scalar triple product $[\vec{U} \vec{V} \vec{W}]$ is