Displacement and position vectors

A vector is a physical quantity which has both magnitude and direction.

The distance moved by an object from one point A to another point B in the direction from A to B is called displacement AB.

In the xy-plane, all vectors having their initial points at the origin and their end points elsewhere are defined as position vectors. Position vectors are named by the coordinates of their end points.

If an object moves from point A to another point say B, there is a displacement.

There are many vector quantities, some of which are: displacement, velocity, acceleration, force, momentum, electric field, and magnetic field.

Other physical quantities have only magnitude; these quantities are called scalars.

Examples of scalars: distance, speed, pressure, time, and temperature.

Normally vectors are named by either two capital letters with an arrow above, e.g., $\vec{OA}, \vec{AB}$, etc. Also a single capital letter or small letter in bold print. E.g., and sometimes a single small letters with a bar below.

Two or more vectors are said to be equivalent if and only if they have the same magnitude and direction.

Consider the following diagram.

If an object moves from point A to another point say B, there is a displacement

$\overrightarrow{AB} \text{ which is the distance from A to B.}$

From the above figure $\overrightarrow{AB}$ is the vector since it has magnitude as well as direction.

There are many vector quantities, some of which are: displacement, velocity, acceleration, force, momentum, electric field, and magnetic field.

Other physical quantities have only magnitude; these quantities are called scalars. For example distance, speed, pressure, time and temperature.

In the x-plane, all vectors with initial points at the origin and their end points elsewhere are called position vectors. Position vectors are named by the coordinates of their end points.

Consider the following diagram.

From Figure above, $\overrightarrow{OA} = (2, 3)$, $\overrightarrow{OB} = (-4, 2)$ and $\overrightarrow{OC} = (3, -4)$ are position vectors of points A, B, and C respectively.

Example 1

Write the position vectors of the following points: (a) A (1, -1) (b) B (-4, -3) (c) C = (u, v) where u and v are any real numbers. Give their horizontal and vertical components.

Solution:

(a) $\vec{OA} = (1,-1)$ Horizontal component = $(1, 0)$ Vertical component = $(0,-1)$

(b) $\vec{OB} = (-4,-3)$ Horizontal component = $(-4, 0)$ Vertical component = $(0,-3)$

(c) $\vec{OC} = (u,v)$ Horizontal component = $(u, 0)$ Vertical component = $(0, v)$

Example 2

For each of vectors a and b shown in the figure below, draw a pair of equivalent vectors.

Solution: The following figure shows the vectors a and b and their respective pairs of equivalent vectors.

The Unit Vectors i and j

Definition: A unit vector is a position vector of unit length in the positive direction of the x-axis or y-axis in the xy-plane.

The letters i and j are used to represent unit vectors in the x-axis and y-axis respectively.

Consider the following sketch.

We can write $\mathbf{i}$ and $\mathbf{j}$ in terms of position vectors as $\mathbf{i} = (1, 0)$ and $\mathbf{j} = (0, 1)$, from the figure above, $\overrightarrow{OP} = (x, y)$

Now $\overrightarrow{OP}$ can be resolved into $\overrightarrow{OP} = (x, 0) + (0, y)$ and factorized into $\overrightarrow{OP} = x \cdot (1, 0) + y \cdot (0, 1)$ But $\mathbf{i} = (1, 0)$ and $\mathbf{j} = (0, 1)$ so:

$\boxed{\overrightarrow{OP} = x(1,0) + y(0,1)}$

Example 3

Write the following vectors in terms of i and j vectors.

(a) $\mathbf{a} = (-3,4)$, (b) $\mathbf{b} = (-5,5)$ (c) $\mathbf{c} = (3,2)$ and (d) $\mathbf{d} = (u, v)$

Solution

(a) $\mathbf{a} = (-3,4) = (-3,0) + (0,-4)$ $= -3(1,0) + -4(0,1)$ $= -3\mathbf{i} + 4\mathbf{j}$ $= -3\mathbf{i} - 4\mathbf{j}$

(b) $\mathbf{b} = (-5,5) = -5\mathbf{i} + 5\mathbf{j}$

(c) $\mathbf{c} = (3,2) = 3\mathbf{i} + 2\mathbf{j}$

(d) $\mathbf{d} = (u,v) = u\mathbf{i} + v\mathbf{j}$

Example 4

Write the following vectors as position vectors.

(a) $S = -8i$ (b) $U = 7j$ (c) $V = \frac{2}{3}i + \frac{1}{6}j$ (d) $D = ui + vj$

Solution

(a) $S = -8i$ $S = -8i + 0j$ $= -8(1, 0) + 0(0, 1)$ $= (-8, 0) + (0, 0)$ $= \underline{(-8, 0)}$

(b) $U = 7j = 0i + 7j$ $U = 0(1, 0) + 7(0, 1)$ $= (0, 0) + (0, 7)$ $= (0, 7)$

(c) $V = \frac{2}{3}(1, 0) + \frac{1}{6}(0, 1)$ $= \frac{2}{3}(1, 0) + \frac{1}{6}(0, 1)$ $= \left(\frac{2}{3}, 0\right) + \left(0, \frac{1}{6}\right)$ $= \left(\frac{2}{3}, \frac{1}{6}\right)$

(d) $D = u\mathbf{i} + v\mathbf{j}$ $= u(1, 0) + (0, v)$ $= (u, 0) + (0, v)$ $= (u, v)$