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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:

$\vec{a}$ = 7.1i + 8.9j
$\vec{b}$ = 5.8i + 2.5j

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

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up vote 1 down vote accepted

Compute $a \cdot b/|b|$ as the component of $a$ along $b$.

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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 $$\frac{63.43}{\sqrt{39.89}}$$

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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.

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