Name: Resolution of Vector Into Components
Uploaded: 2019-07-19
Duration: 8 min 23 s
Description: The resolution of a vector means splitting of a vector into two or more directions. Let us learn how to resolve a vector through this video.

Question

The position vectors of the point

P, Q, R, S

are

\hat{i} + \hat{j} + \hat{k}, 2 \hat{i} + 5 \hat{j}, 3 \hat{i} + 2 \hat{j} - 3 \hat{k}

and

\hat{i} - 6 \hat{j} - \hat{k}

respectively. Prove that the line

PQ

and

RS

are parallel.

Accepted Answer

Given that, the position vectors of the given points $P, Q, R, S$ are

$\vec{OP} = \hat{i} + \hat{j} + \hat{k}$

$\vec{OQ} = 2 \hat{i} + 5 \hat{j}$

$\vec{OQ} = 2 \hat{i} + 5 \hat{j}$

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

$\vec{OS} = \hat{i} - 6 \hat{j} - \hat{k}$

Now, $\vec{PQ} = \vec{OQ} - \vec{OP}$

$= (2 \hat{i} + 5 \hat{j}) - (\hat{i} + \hat{j} + \hat{k})$

$= 2 \hat{i} + 5 \hat{j} - \hat{i} - \hat{j} - \hat{k}$

So, $\vec{PQ} = \hat{i} + 4 \hat{j} - \hat{k}$

Consider, $\vec{RS} = \vec{OS} - \vec{OR}$

$= (\hat{i} - 6 \hat{j} - \hat{k}) - (3 \hat{i} + 2 \hat{j} - 3 \hat{k})$

$= - 2 \hat{i} - 8 \hat{j} + 2 \hat{k}$

$= - 2 (\hat{i} + 4 \hat{j} - \hat{k})$

$∴ \vec{RS} = - 2 \vec{PQ}$

We know that, two vectors $\vec{a}$ and $\vec{b}$ are parallel vectors if $\vec{a} = λ \vec{b}$ where $λ$ is a scalar.

$∴ \vec{RS}$ and $\vec{PQ}$ are parallel vectors.

The position vectors of the point $P, Q, R, S$ are $\hat{i} + \hat{j} + \hat{k}, 2 \hat{i} + 5 \hat{j}, 3 \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $\hat{i} - 6 \hat{j} - \hat{k}$ respectively. Prove that the line $PQ$ and $RS$ are parallel.

Chapter-Wise 3D Mathematics Videos from Tamil Nadu Board Mathematics Standard 11 Vol II

Resolution of Vector Into Components

Introduction to Scalars & Vectors

Scalar Product

Learn and Prepare for any exam you want !

The position vectors of the point P, Q, R, S are i^+j^+k^, 2i^+5j^, 3i^+2j^-3k^ and i^-6j^-k^ respectively. Prove that the line PQ and RS are parallel.