Find the component of $\vec{a}$ along $\vec{b}$?

Question

I'm trying to do homework for my physics class, and it says I should find 'the component of $\vec{a}$ along the direction of $\vec{b}$'. The vectors are:

$\vec{a} = 7.1\hat i + 8.97 \hat j $

$\vec{b} = 5.8\hat i + 2.5\hat j$

I know how to find the $x$ and $y$ components but I've never done this before. How do I do it?

Answer 1

Compute $\dfrac{\vec a \cdot \vec b}{|b|}$ as the component of $\vec a$ along $\vec b$.

Answer 2

Hint: the component of $a$ along $b$ (also known as the scalar projection of $a$ onto $b$) is given by

$$\text{comp}_b a = \frac{a \cdot b}{\vert b \vert}$$

where $a \cdot b$ is the dot product.

Answer 3

Using the formula

$$\text{comp}_b a = \frac{a \cdot b}{\vert b \vert}$$

with the given vectors

$$\vec{a} = (7.1, 8.9)$$ $$\vec{b} = (5.8,2.5)$$ we get that

$$a\cdot b = (7.1, 8.9)\cdot (5.8, 2.5) = 7.1\cdot 5.8 + 8.9\cdot 2.5 = 63.43$$

Then $$\vert b\vert = \sqrt{{5.8}^{2} + 2.5^{2}} = \sqrt{33.64 + 6.25}= \sqrt{39.89}$$

Therefore the answer is $$\frac{63.43}{\sqrt{39.89}}$$