CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Perpendicular Distance of a Point from a Line

Find the vect...

Question

Find the vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $r a = \land i + 2 \land j - 3 \land k$ and $1 b = 3 \land i - \land j + 2 \land k$ .

A

ˆ i - 11 ˆ j - 7 ˆ k

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

B

2 ˆ i + 11 ˆ j - 3 ˆ k

No worries! We‘ve got your back. Try BYJU‘S free classes today!

C

ˆ i - 5 ˆ j + 7 ˆ k

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D

3 ˆ i - 18 ˆ j - 4 ˆ k

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $ˆ i - 11 ˆ j - 7 ˆ k$
$\begin{matrix} let \to a & = i + 2 j - 3^k \to b & = 3^i -^j + 2^k \to a \times \to b & = (^i) (4 - 3) -^j (2 + 9) + k (- 1 - 6) =^i - 11 j - 7^k \end{matrix}$
$\begin{matrix} | \to a | \times \to b ∣ & = \sqrt{(1)^{2} + (- 1)^{2} + (- 7)^{2}} = \sqrt{171} \end{matrix}$
Let $\to p$ be unit vector Perpendicular $t o \cdot \to a \cdot \to b$
$\begin{matrix} \to p & = \frac{1}{| \to a \times \to b |} [\to a \times \to b] = \frac{[^i - 11^j - 7^k]}{\sqrt{171}} \end{matrix}$
The required vector Perpendicular to $\to a a n d \to b$ whose magnitude is $\sqrt{171}$
$\begin{matrix} \to x & = \sqrt{171} \times \frac{[i - 11^j - 7^k]}{\sqrt{171}} =^ı - 11^j - 7^k \end{matrix}$

Suggest Corrections

thumbs-up

0

Similar questions

\vec{a} = \hat{i} + 4 \hat{j} + 2 \hat{k}, \vec{b} = 3 \hat{i} - 2 \hat{j} + 7 \hat{k} and \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .

\vec{d}

which is perpendicular to both

\vec{a} and \vec{b}

and \vec{c} \cdot \vec{d} = 15 .

\vec{a} = 4 \hat{i} + 5 \hat{j} - \hat{k}, \vec{b} = \hat{i} - 4 \hat{j} + 5 \hat{k} and \vec{c} = 3 \hat{i} + \hat{j} - \hat{k}

. Find a vector

\vec{d}

which is perpendicular to both

\vec{c} and \vec{b} and \vec{d} . \vec{a} = 21

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. Prove that the following vectors are non-coplanar:
(i)

3 \hat{i} + \hat{j} - \hat{k}, 2 \hat{i} - \hat{j} + 7 \hat{k} and 7 \hat{i} - \hat{j} + 23 \hat{k}

\hat{i} + 2 \hat{j} + 3 \hat{k}, 2 \hat{i} + \hat{j} + 3 \hat{k} and \hat{i} + \hat{j} + \hat{k}

Q. Find a vector of magnitude

\sqrt{171}

which is perpendicular to both of the vectors

\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}

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

Find a vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $\to a = ˆ i + 2 ˆ j - 3 ˆ k$ and $\to b = 3 ˆ i - ˆ j + 2 ˆ k$ .

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Triangle Construction 2

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Perpendicular Distance of a Point from a Line

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon