The scalar product

Vectors, grade 12 (5 units) Lesson 12 of 23Learning

The scalar product

The definition

The scalar product of two vectors is a number:

\displaystyle \underline{u}\cdot\underline{v}

\displaystyle =

\displaystyle |\underline{u}|\cdot|\underline{v}|\cdot\cos\alpha

$\displaystyle \alpha$ is the angle between the vectors when they leave the same point, $\displaystyle 0^{\circ}\le\alpha\le180^{\circ}$ .
The dot on the left is an operation between vectors; on the right – a product of numbers.
For example: $\displaystyle {|\underline{u}|}={3}$ , $\displaystyle {|\underline{v}|}={2}$ , $\displaystyle {\alpha}={60^{\circ}}$ : $\displaystyle {\underline{u}\cdot\underline{v}}={3\cdot2\cdot\frac{1}{2}}={3}$ .

The sign and the angle

The lengths are positive, so the sign of $\displaystyle \underline{u}\cdot\underline{v}$ is the sign of $\displaystyle \cos\alpha$ :

$\displaystyle \underline{u}\cdot\underline{v}>0$ – the angle is acute;
$\displaystyle \underline{u}\cdot\underline{v}<0$ – the angle is obtuse;
$\displaystyle {\underline{u}\cdot\underline{v}}={0}$ – the angle is right, or one of the vectors is $\displaystyle \underline{0}$ .

\displaystyle \underline{u}\perp\underline{v}\iff\underline{u}\cdot\underline{v}

\displaystyle =

\displaystyle 0

When the product is known, the angle comes from $\displaystyle {\cos\alpha}={\frac{\underline{u}\cdot\underline{v}}{|\underline{u}|\cdot|\underline{v}|}}$ .

The laws of the product

$\displaystyle {\underline{u}\cdot\underline{v}}={\underline{v}\cdot\underline{u}}$ ;
$\displaystyle {(t\underline{u})\cdot\underline{v}}={t(\underline{u}\cdot\underline{v})}$ ;
$\displaystyle {\underline{u}\cdot(\underline{v}+\underline{w})}={\underline{u}\cdot\underline{v}+\underline{u}\cdot\underline{w}}$ ;
$\displaystyle {\underline{u}\cdot\underline{u}}={|\underline{u}|^2}$ , since the angle between a vector and itself is $\displaystyle 0^{\circ}$ . We write $\displaystyle \underline{u}^2$ .

So brackets are opened as in algebra, and the short multiplication formulas still hold:

\displaystyle (\underline{u}+\underline{v})^2

\displaystyle =

\displaystyle \underline{u}^2+2\underline{u}\cdot\underline{v}+\underline{v}^2

and $\displaystyle {(\underline{u}+\underline{v})\cdot(\underline{u}-\underline{v})}={\underline{u}^2-\underline{v}^2}$ . This is how a length is found: $\displaystyle {|\underline{w}|}={\sqrt{\underline{w}^2}}$ .

Example: a length and an angle

$\displaystyle {|\underline{u}|}={3}$ , $\displaystyle {|\underline{v}|}={2}$ , and the angle between them is $\displaystyle 60^{\circ}$ .
Find $\displaystyle |\underline{u}+\underline{v}|$ and the angle between $\displaystyle \underline{u}$ and $\displaystyle \underline{u}+\underline{v}$ .

The solution below first finds $\displaystyle \underline{u}\cdot\underline{v}$ , and then opens brackets.

$\displaystyle {\underline{u}\cdot\underline{v}}={3\cdot2\cdot\cos60^{\circ}}={3}$ .
$\displaystyle {(\underline{u}+\underline{v})^2}={\underline{u}^2+2\underline{u}\cdot\underline{v}+\underline{v}^2}={9+6+4}={19}$ , so $\displaystyle {|\underline{u}+\underline{v}|}={\sqrt{19}}$ .
$\displaystyle {\underline{u}\cdot(\underline{u}+\underline{v})}={\underline{u}^2+\underline{u}\cdot\underline{v}}={9+3}={12}$ .
$\displaystyle {\cos\beta}={\frac{12}{3\sqrt{19}}}$ , so $\displaystyle \beta\approx23.41^{\circ}$ .

Exercise 1

$\displaystyle {|\underline{u}|}={4}$ , $\displaystyle {|\underline{v}|}={5}$ , and the angle between them is $\displaystyle 120^{\circ}$ .

\displaystyle \underline{u}\cdot\underline{v}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle |\underline{u}+\underline{v}|

, exactly

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Open the brackets

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle (2\underline{u}-\underline{v})\cdot(\underline{u}+3\underline{v})

– put in the numbers

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

$\displaystyle {|\underline{u}|}={|\underline{v}|}={1}$ , and $\displaystyle {|\underline{u}+\underline{v}|}={\sqrt{3}}$ .

\displaystyle \underline{u}\cdot\underline{v}

(hint:

\displaystyle {(\underline{u}+\underline{v})^2}={3}

)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The angle between

\displaystyle \underline{u}

and

\displaystyle \underline{v}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

$\displaystyle {|\underline{u}|}={2}$ , $\displaystyle {|\underline{v}|}={3}$ , and the angle between them is $\displaystyle 60^{\circ}$ .
Find $\displaystyle t$ for which $\displaystyle \underline{u}+t\underline{v}$ is perpendicular to $\displaystyle \underline{u}-\underline{v}$ . (Hint: open $\displaystyle (\underline{u}+t\underline{v})\cdot(\underline{u}-\underline{v})$ , put in the numbers, and set it equal to 0.)

Exercise 4

True or false? Explain.

If $\displaystyle {\underline{u}\cdot\underline{v}}={\underline{u}\cdot\underline{w}}$ and $\displaystyle \underline{u}\neq\underline{0}$ , then $\displaystyle {\underline{v}}={\underline{w}}$ .
If $\displaystyle {|\underline{u}+\underline{v}|}={|\underline{u}-\underline{v}|}$ , then $\displaystyle \underline{u}\perp\underline{v}$ .

Formulas: vectors, grade 12 (5 units)

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Solids in space

The volume of a prism and a cylinder – $\displaystyle S$ the base’s area, $\displaystyle h$ the solid’s height
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\cdot h$
The volume of a pyramid and a cone – $\displaystyle S$ the base’s area, $\displaystyle h$ the solid’s height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\cdot h}{3}$
The lateral area of a right cone – $\displaystyle R$ the base’s radius, $\displaystyle \ell$ the slant height
$\displaystyle M$ $\displaystyle =$ $\displaystyle \pi R\ell$
The volume of a sphere – $\displaystyle R$ its radius
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4}{3}\pi R^3$
The surface area of a sphere
$\displaystyle M$ $\displaystyle =$ $\displaystyle 4\pi R^2$
The diagonal of a box of dimensions $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2+c^2}$

Trigonometry in the triangle

The law of sines – $\displaystyle R$ the radius of the circumscribed circle
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$
The area of a triangle – two sides and the angle between them
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

Vectors in the geometric approach

The triangle rule
$\displaystyle \overrightarrow{AB}+\overrightarrow{BC}$ $\displaystyle =$ $\displaystyle \overrightarrow{AC}$
A vector between two points – the end minus the start
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle \overrightarrow{OB}-\overrightarrow{OA}$
Collinear (parallel) vectors
$\displaystyle \underline{v}=t\cdot\underline{u}$
The point $\displaystyle M$ divides the segment $\displaystyle AB$ in the ratio $\displaystyle m:n$
$\displaystyle \overrightarrow{OM}=\frac{n}{m+n}\overrightarrow{OA}+\frac{m}{m+n}\overrightarrow{OB}$ $\displaystyle AM:MB=m:n$
$\displaystyle M$ the midpoint of the segment $\displaystyle AB$
$\displaystyle \overrightarrow{OM}=\frac{1}{2}\left(\overrightarrow{OA}+\overrightarrow{OB}\right)$

The scalar product

The length of a vector $\displaystyle {\underline{u}}={(u_1,u_2,u_3)}$
$\displaystyle |\underline{u}|=\sqrt{\underline{u}\cdot\underline{u}}=\sqrt{u_1^2+u_2^2+u_3^2}$
The scalar product in coordinates
$\displaystyle \underline{u}\cdot\underline{v}$ $\displaystyle =$ $\displaystyle u_1v_1+u_2v_2+u_3v_3$
The scalar product – $\displaystyle \alpha$ the angle between the vectors
$\displaystyle \underline{u}\cdot\underline{v}$ $\displaystyle =$ $\displaystyle |\underline{u}|\cdot|\underline{v}|\cdot\cos\alpha$
Perpendicular vectors
$\displaystyle \underline{u}\perp\underline{v}\iff\underline{u}\cdot\underline{v}=0$
A short scalar product
$\displaystyle (\underline{u}+\underline{v})^2$ $\displaystyle =$ $\displaystyle \underline{u}^2+2\underline{u}\cdot\underline{v}+\underline{v}^2$

Points and vectors in axes

A vector between two points
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)$
The distance between two points
$\displaystyle AB$ $\displaystyle =$ $\displaystyle \sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}$
The midpoint of the segment $\displaystyle AB$
$\displaystyle M=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2},\frac{z_A+z_B}{2}\right)$

The line and the plane

A line – $\displaystyle \underline{v}$ a point on the line, $\displaystyle \underline{u}$ the direction vector
$\displaystyle \underline{x}=\underline{v}+t\cdot\underline{u}$
A plane in parametric form – $\displaystyle \underline{u}$ , $\displaystyle \underline{v}$ not parallel
$\displaystyle \underline{x}=\underline{p}+t\cdot\underline{u}+s\cdot\underline{v}$
A plane’s equation – $\displaystyle {\underline{n}}={(a,b,c)}$ the normal vector
$\displaystyle ax+by+cz+d=0$
The plane through $\displaystyle P(x_1,y_1,z_1)$ whose normal vector is $\displaystyle (a,b,c)$
$\displaystyle a(x-x_1)+b(y-y_1)+c(z-z_1)=0$

Distances

The distance of a point $\displaystyle P(x_1,y_1,z_1)$ from the plane $\displaystyle {ax+by+cz+d}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|a\cdot x_1+b\cdot y_1+c\cdot z_1+d|}{\sqrt{a^2+b^2+c^2}}$
The distance between the parallel planes $\displaystyle {ax+by+cz+d_1}={0}$ and $\displaystyle {ax+by+cz+d_2}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}$

Angles

The angle between two lines – $\displaystyle \underline{u}$ , $\displaystyle \underline{v}$ the direction vectors
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{|\underline{u}\cdot\underline{v}|}{|\underline{u}|\cdot|\underline{v}|}$
The angle between the line $\displaystyle {\underline{x}}={\underline{v}+t\cdot\underline{u}}$ and the plane whose normal is $\displaystyle \underline{n}$
$\displaystyle \sin\beta$ $\displaystyle =$ $\displaystyle \frac{|\underline{n}\cdot\underline{u}|}{|\underline{n}|\cdot|\underline{u}|}$
The angle between two planes – $\displaystyle \underline{n_1}$ , $\displaystyle \underline{n_2}$ the normals
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{|\underline{n_1}\cdot\underline{n_2}|}{|\underline{n_1}|\cdot|\underline{n_2}|}$