Question

Given two vector A=3i+4j and B=i+j. Theta is the angle between A and B. Find the component of A along B and component of A perpendicular to B.

Answer 1

please see the attachment

Answer 2

Answer:

The component of A along B and component of A perpendicular to B are $\dfrac{7}{\sqrt{2}}$ and $\dfrac{1}{\sqrt{2}}$.

Explanation:

Given that,

Two vectors

$A = 3i+4j$

$B=i+j$

We need to calculate the component of A along B

$A\ along\ B=\dfrac{\vec{A}\cdot\vec{B}}{|\vec{B}|}$

$A\ along\ B=\dfrac{(3i+4j)\cdot(i+j)}{\sqrt{2}}$

$A\ along\ B=\dfrac{7}{\sqrt{2}}$

We need to calculate the component vector of A perpendicular to B

$A\ perpendicular\ to B=\vec{A}-\dfrac{\vec{A}\cdot\vec{B}}{}\vec{B}$

$A\ perpendicular\ to B=(3i+4j)-\dfrac{7}{2}(i+j)$

$A\ perpendicular\ to B=\dfrac{-i+j}{2}$

Now, the component of A perpendicular to B

$|A\ perpendicular\ to B|=|\dfrac{-i+j}{2}|$

$|A\ perpendicular\ to B|=\dfrac{1}{\sqrt{2}}$

Hence, The component of A along B and component of A perpendicular to B are $\dfrac{7}{\sqrt{2}}$ and $\dfrac{1}{\sqrt{2}}$.