Name: Cross Product
Uploaded: 2019-07-19
Duration: 4 min 16 s
Description: The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a particular direction and magnitude. Watch this vi...

Question

Express the vector $\vec{a} = (6 \hat{i} - 3 \hat{j} - 6 \hat{k})$ as sum of two vectors such that one is parallel to the vector $\vec{b} = (\hat{i} + \hat{j} + \hat{k})$ and the other is perpendicular to $\vec{b}$ .

Accepted Answer

Given, $\vec{a} = (6 \hat{i} - 3 \hat{j} - 6 \hat{k})$ and $\vec{b} = (\hat{i} + \hat{j} + \hat{k})$

Let $\vec{c}$ is the vector that is parallel to $\vec{b}$ and $\vec{d}$ is perpendicular to $\vec{b}$ .

So, $\vec{c} = λ \vec{b}$ & $\vec{b} \cdot \vec{d} = 0$

$∴ \vec{a} = \vec{c} + \vec{d}$

Now, $\vec{b} \cdot \vec{a} = \vec{b} \cdot (\vec{c} + \vec{d})$

$\Rightarrow (\hat{i} + \hat{j} + \hat{k}) \cdot (6 \hat{i} - 3 \hat{j} - 6 \hat{k}) = \vec{b} \cdot λ \vec{b} + \vec{b} \cdot \vec{d}$

$\Rightarrow 6 - 3 - 6 = λ (| \vec{b} |^{2}) + 0$ $[∵ \vec{b} \cdot \vec{b} = {|\vec{b}|}^{2}]$

$\Rightarrow - 3 = 3 λ$ $[∵ |\vec{b}| = \sqrt{1^{2} + 1^{2} + 1^{2}} = \sqrt{3}]$

$\Rightarrow λ = - 1$

Now, $\vec{c} = λ \vec{b} = - 1 (\hat{i} + \hat{j} + \hat{k}) = - (\hat{i} + \hat{j} + \hat{k})$

$∴ \vec{a} = \vec{c} + \vec{d}$

$\Rightarrow (6 \hat{i} - 3 \hat{j} - 6 \hat{k}) = - (\hat{i} + \hat{j} + \hat{k}) + \vec{d}$

$\Rightarrow \vec{d} = 7 \hat{i} - 2 \hat{j} - 5 \hat{k}$

Consider, $\vec{a} = \vec{c} + \vec{d}$

$∴ \vec{a} = - (\hat{i} + \hat{j} + \hat{k}) + (7 \hat{i} - 2 \hat{j} - 5 \hat{k})$ .

Express the vector $\vec{a} = (6 \hat{i} - 3 \hat{j} - 6 \hat{k})$ as sum of two vectors such that one is parallel to the vector $\vec{b} = (\hat{i} + \hat{j} + \hat{k})$ and the other is perpendicular to $\vec{b}$ .

Important Questions on Scalars or Dot, Product of Vectors

Learn and Prepare for any exam you want !

Express the vector a→=(6i^-3j^-6k^) as sum of two vectors such that one is parallel to the vector b→=(i^+j^+k^) and the other is perpendicular to b→.

Important Questions on Scalars or Dot, Product of Vectors

Learn and Prepare for any exam you want !

Express the vector $\vec{a} = (6 \hat{i} - 3 \hat{j} - 6 \hat{k})$ as sum of two vectors such that one is parallel to the vector $\vec{b} = (\hat{i} + \hat{j} + \hat{k})$ and the other is perpendicular to $\vec{b}$ .