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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} = 7.1\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?

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4 Answers 4

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

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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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I believe the component of A along B must be a vector. The previous answer gives the length of the component of A along B. Now that must be multiplied by a unit vector in the direction of B. So my answer would be:

$comp_{b}A = \frac {A \cdot B}{|B|}$ multiplying by the vector $\frac {B}{|B|}$ we get $\frac {A \cdot B}{|B||B|} B = \frac {A \cdot B}{B \cdot B} B$

Then to get the component of A perpendicular to B, you subtract that from A.

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    $\begingroup$ Precise answer ! $\endgroup$
    – Vicrobot
    Commented Jul 14, 2020 at 18:26
  • $\begingroup$ I think that "component" can be interpreted either as vector component (projection) or scalar component (scalar projection). The latter, which the other answers are all giving, can be negative, so isn't actually a length. $\endgroup$
    – ryang
    Commented May 31 at 15:48
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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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