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Introductory vectors

Covers definition of a vector, basic algebra of vectors, calculation of magnitude, angle

Basic properties of vectors

A vector is a quantity with both magnitude and direction. 

Relative to a cartesian frame of reference, $(x,y)$ or $(x,y,z)$, a vector is often expressed in terms of components;

$$ \left(\begin{array}{c}a \\ b\end{array}\right),\; \left(\begin{array}{c}a \\ b \\ c \end{array}\right) $$

The same vectors can also be expressed in terms of unit vectors, $\bf{i},\bf{j},\bf{k}$ that are parallel to the coordinate axes; $$ \left(\begin{array}{c}a \\ b\end{array}\right) = a{\bf{i}} + b{\bf{j}} $$ $$ \left(\begin{array}{c}a \\ b\\c \end{array}\right) = a{\bf{i}} + b{\bf{j}} + c{\bf{k}} $$


The length of a vector ${\bf V} = a{\bf i} + b{\bf j} + c{\bf k}$ is given by $$ |{\bf V}| = \sqrt{a^2 + b^2 + c^2} $$

The unit vector in the parallel to ${\bf V}$ is given by $$ {\bf\widehat{V}} = \frac{\bf V}{\left|\bf{V}\right|} $$


Algebra of vectors

Vectors form a vector space, that is, addition of vectors and multiplication by scalars are performed as expected:

$$\left(\begin{array}{c}a \\ b \\ c \end{array}\right) +  \left(\begin{array}{c}p \\ q \\ r \end{array}\right) =   \left(\begin{array}{c}a+p \\b+ q \\c+ r \end{array}\right) $$

$$\lambda \left(\begin{array}{c}a \\ b \\ c \end{array}\right) = \left(\begin{array}{c}\lambda a \\ \lambda b \\ \lambda c \end{array}\right) $$


The dot product

Given two non-zero vectors, ${\bf V} = a{\bf i} + b{\bf j} + c{\bf k}$ and ${\bf W} = p{\bf i} + q{\bf j} + r{\bf k}$, the vector dot product is defined as $$ {\bf V} \cdot {\bf W} = ap + bq + cr = \left|{\bf V}\right|\left|{\bf W}\right|\cos \theta $$ where $\theta$ is the angle between the two vectors.

Basic vectors 1

Determine the angle between the two vectors $$ {\bf V} = 3{\bf i} + 2{\bf j} + 2{\bf k},\;{\bf W} = -1{\bf i} + 3{\bf j} - 2\bf{k}, $$
solution - press button to display

$$|{\bf V}|= \sqrt{3^2 + 2^2 + 2^2} = \sqrt{17} $$

$$ |{\bf W}|= \sqrt{(-1)^2 + 3^2 + (-2)^2} = \sqrt{14} $$

$${\bf V}\cdot {\bf W} = (3)(-1) + (2)(3) + (2)(-2) = -1$$

We know that $$ \cos \theta = \frac{{\bf V}\cdot{\bf W}}{|{\bf V}||{\bf W}|} $$ and hence $$ \cos \theta = \frac{-1}{\sqrt{17}\sqrt{14}} = 93.72^\circ $$

Basic vectors 2

Determine the value of $\alpha$ such that the following vectors are perpendicular $$ {\bf V} = 2{\bf i} + 3{\bf j} -4{\bf k}, \; {\bf W} = 3{\bf i} + \alpha {\bf j} + 2{\bf k} $$
solution - press button to display
Two vectors are perpendicular if their dot product is zero. $$ {\bf V}\cdot {\bf W} = 6 +3\alpha -8 \Rightarrow -2+3\alpha =0 \Rightarrow \alpha = \frac{2}{3} $$