Uses of the Scalar Product

This revision note covers several applications of the scalar product for vectors – namely, how you can use the scalar product to:

Recall that a formula for the scalar (or ‘dot’) between vectors $a$ and $b$ is

$a \cdot b = |a| |b| \cos θ$

To find the angle between two lines, find the angle between their direction vectors

- For example, if the lines have equations $r = a_{1} + s d_{1}$ and $r = a_{2} + t d_{2}$ , then the angle $θ$ between the lines is given by

$θ = \cos^{- 1} (\frac{d_{1} \cdot d_{2}}{|d_{1}| |d_{2}|})$

$(2 i - 3 j + 5 k) \cdot (- 4 i - j + k) = 2 \times (- 4) + (- 3) \times (- 1) + 5 \times 1 = - 8 + 3 + 5 = 0$

Suppose that we have a line $l$ with equation $r = a + t d$ and a point $P$ not on $l$
Let $F$ be the point on $l$ which is closest to $P$ (sometimes called the foot of the perpendicular)
- Then the line between $F$ and $P$ will be perpendicular to the line $l$
To find the closest point $F$
- Call $f = \vec{OF}$ and $p = \vec{O P}$
- Since $F$ lies on $l$ , we have $f = a + t_{0} d$ , for a unique real number $t_{0}$
- Find the vector $\vec{F P}$ using $p - f$
- $\vec{F P}$ is perpendicular to $d$ so form an equation using $(p - f) \cdot d = 0$
- Solve this equation to find the value of $t_{0}$
- Use the value of $t_{0}$ to find $f$
The shortest distance between the point and the line is the length
Note that the shortest distance between the point and the line is sometimes referred to as the length of the perpendicular

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It can be easier and clearer to work with column vectors when dealing with scalar products.

Uses of the Scalar Product (CIE A Level Maths: Pure 3)