Conversion of vectors to scalars in rectilinear motion

Resolution

$\overrightarrow{r}=3·\overrightarrow{i} m$

It represents the position vector of the motion. Remember that a vector has magnitude and direction. In this case:

• Magnitude: 3
• Direction: same as the unit vector $\overrightarrow{i}$ 

Movements that take place in the direction of the unit vector $\overrightarrow{i}$ are those associated with the x axis, therefore:

$$\boxed{\overrightarrow{r}=3·\overrightarrow{i}m\Rightarrow x=3 m}$$ 

$∆\overrightarrow{r}=-3·\overrightarrow{j}m$

It is the displacement vector of the motion:

• Magnitude: 3 (magnitude is never negative)
• Direction: That given by the unit vector $-\overrightarrow{j}$ 

Movements that take place in the direction of the unit vector $\overrightarrow{j}$ are those associated to the y-axis, therefore:

$$\boxed{∆\overrightarrow{r}=-3·\overrightarrow{j}m\Rightarrow ∆y=-3 m}$$

$\overrightarrow{v}=4·(-\overrightarrow{j}) m/s$ 

It is normally written $\overrightarrow{v}=4·(-\overrightarrow{j})=-4·\overrightarrow{j}$ .

Which is the velocity vector of the motion. Following a similar reasoning to the previous one:

$$\boxed{\overrightarrow{v}=4·(-\overrightarrow{j}) m/s=-4·\overrightarrow{j}m/s\Rightarrow v_y=-4 m/s}$$

$\overrightarrow{v}=3·t·\overrightarrow{i}m/s$

In this case the velocity vector of the motion depends on time. We can write:

• Magnitude 3·t
• Direction: given by the unit vector $\overrightarrow{i}$ 

The vector is associated to x-axis, therefore:

$$\boxed{\overrightarrow{v}=3·t·\overrightarrow{i}m/s\Rightarrow v_x=3·t m/s}$$