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Integration of Piecewise Continuous Functions

Find a vector...

Question

Find a vector of magnitude $\sqrt{2}$ units and coplanar with vectors $3 i - j - k$ and $i + j - 2 k$ and perpendicular to vector $2 i + 2 j + k$ .

A

r = \pm \frac{1}{5} (- 3 i + 5 j - 4 k)

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B

r = \pm \frac{1}{5} (3 i - 5 j + 4 k)

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C

r = \pm \frac{1}{4} (2 i - 2 j + 4 k)

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D

r = \pm \frac{1}{4} (- 2 i + 2 j - 4 k)

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Solution

The correct option is B $r = \pm \frac{1}{5} (3 i - 5 j + 4 k)$
$r = t {a \times (b \times c)}$ represents a vector coplanar with $b$ and $c$ and perpendicular to $a$ .
$r = t [(a \cdot c) b - (a \cdot b) c]$
Substitute the values of $a . c$ and $a . b$
$∴ r = t (2 b - 3 c) = t [2 (3 i - j - k) - 3 (i + j - 2 k)]$
$\Rightarrow r = t (3 i - 5 j + 4 k) .$
Take mod of both sides. $| r | = | t | \sqrt{9 + 25 + 16}$
$\Rightarrow \sqrt{2} = | t | .5 \sqrt{2}$ $| t | = \frac{1}{5} ∴ t = \pm \frac{1}{5}$
$∴$ Required vector is $\pm \frac{1}{5} (3 i - 5 j + 4 k)$

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Similar questions

Q. Find the shortest distance between the following pairs of lines whose vector equations are:

(i)

\vec{r} = 3 \hat{i} + 8 \hat{j} + 3 \hat{k} + λ (3 \hat{i} - \hat{j} + \hat{k}) and \vec{r} = - 3 \hat{i} - 7 \hat{j} + 6 \hat{k} + μ (- 3 \hat{i} + 2 \hat{j} + 4 \hat{k})

\vec{r} = (3 \hat{i} + 5 \hat{j} + 7 \hat{k}) + λ (\hat{i} - 2 \hat{j} + 7 \hat{k}) and \vec{r} = - \hat{i} - \hat{j} - \hat{k} + μ (7 \hat{i} - 6 \hat{j} + \hat{k})

\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) and \vec{r} = (2 \hat{i} + 4 \hat{j} + 5 \hat{k}) + μ (3 \hat{i} + 4 \hat{j} + 5 \hat{k})

\vec{r} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - t) \hat{k} and \vec{r} = (s + 1) \hat{i} + (2 s - 1) \hat{j} - (2 s + 1) \hat{k}

\vec{r} = (λ - 1) \hat{i} + (λ + 1) \hat{j} - (1 + λ) \hat{k} and \vec{r} = (1 - μ) \hat{i} + (2 μ - 1) \hat{j} + (μ + 2) \hat{k}

\vec{r} = (2 \hat{i} - \hat{j} - \hat{k}) + λ (2 \hat{i} - 5 \hat{j} + 2 \hat{k}) and, \vec{r} = (\hat{i} + 2 \hat{j} + \hat{k}) + μ (\hat{i} - \hat{j} + \hat{k})

\vec{r} = (\hat{i} + \hat{j}) + λ (2 \hat{i} - \hat{j} + \hat{k}) and, \vec{r} = 2 \hat{i} + \hat{j} - \hat{k} + μ (3 \hat{i} - 5 \hat{j} + 2 \hat{k})

\vec{r} = (8 + 3 λ) \hat{i} - (9 + 16 λ) \hat{j} + (10 + 7 λ) \hat{k}

\vec{r} = 15 \hat{i} + 29 \hat{j} + 5 \hat{k} + μ (3 \hat{i} + 8 \hat{j} - 5 \hat{k})

[NCERT EXEMPLAR]

Q. Find the angle between the following pairs of lines:
(i)

\vec{r} = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 2 \hat{k}) and \vec{r} = \hat{i} - \hat{j} + 2 \hat{k} - μ (2 \hat{i} + 4 \hat{j} - 4 \hat{k})

\vec{r} = (3 \hat{i} + 2 \hat{j} - 4 \hat{k}) + λ (\hat{i} + 2 \hat{j} + 2 \hat{k}) and \vec{r} = (5 \hat{j} - 2 \hat{k}) + μ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})

\vec{r} = λ (\hat{i} + \hat{j} + 2 \hat{k}) and \vec{r} = 2 \hat{j} + μ \{(\sqrt{3} - 1) \hat{i} - (\sqrt{3} + 1) \hat{j} + 4 \hat{k}\}

2 ¯ i - 3 ¯ j + 4 ¯ ¯ ¯ k, ¯ i - 2 ¯ j + 3 ¯ ¯ ¯ k

3 ¯ i + ¯ j - 2 ¯ ¯ ¯ k

Q. If the points whose position vectors are

3 ¯ i - 2 ¯ j - ¯ ¯ ¯ k, 2 ¯ i + 3 ¯ j - 4 ¯ ¯ ¯ k, - ¯ i + ¯ j + 2 ¯ ¯ ¯ k

4 ¯ i + 5 ¯ j + λ ¯ ¯ ¯ k

are coplanar then show that

λ = \frac{- 146}{m}

m

Q. The angle between the vectors

3 i + j + 2 k

2 i - 2 j + 4 k

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