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Definition of Vector

The component...

Question

The component of a vector $\to a$ along and perpendicular to a nonzero vector $\to b$ is:

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Solution

$- - \to O A = \to a$ and $M$ is the foot of perpendicular on $O B$ .
The scalar component of $\to a$ along $\to b = O M$
$= ∣ ∣ \to a ∣ ∣ cos θ$
$= \to a . \frac{\to b}{∣ ∣ ∣ \to b ∣ ∣ ∣}$
The scalar component of $\to a$ perpendicular to $\to b = A M$
$= ∣ ∣ \to a ∣ ∣ sin θ$
$= \frac{∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣}{∣ ∣ ∣ \to b ∣ ∣ ∣}$
The vector component of $\to a$ along $\to b = - - \to O M$
$= O M \frac{\to b}{∣ ∣ ∣ \to b ∣ ∣ ∣}$
$= \frac{(\to a . \to b) \to b}{\to b . \to b}$
The vector component of $\to a$ perpendicular to $\to b$
$\Rightarrow - - \to M A = - - \to O A - - - \to O M$
$= \to a - \frac{(\to a . \to b) \to b}{(\to b . \to b)}$
$= \frac{\to b \times (\to a \times \to b)}{\to b . \to b}$

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