Scalar

Scalar quantities like volume, density, speed, energy, mass, and time, only have the “size” (magnitude) not direction.

Some physical quantities can be expressed by a number and its corresponding unit. A scalar is a quantity that has a magnitude but no direction.

Vector

Vector, however, have both magnitude and direction. For example, force and acceleration are one of the few that need both the “size”(newton and speed)

Geometrical Presentation

figure 1

A vector is drawn geometrically by a directed line segment.

Characteristic of Vector

• Initial point (tail)- the point where the vector begins. —{A
• Terminal point(tip)- the point where the vector ends. —{B

Denotation

We can denote vectors by the initial point with the terminal point. In figure one, we will denote the vector as $\overline {AB}$ Or the denotation of the full name, in this case, the vector can be denote as $a\text{ or } \vec a$.

Standard Form

If the initial point is at the origin, the basic requirement of standard form is meet.

Direction

The direction of the vector is determine by the angle between the vector and the positive x axis.

Magnitude (modulus-complex #)

The length of the line represents the magnitude of the vector.

<aside> 💡 Seems familiar? The polar form of complex number have some in common of vector.

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Navigational Direction

Similar to standard form, the initial point is at the center of the “compass”

Direction

Unlike standard form, the direction of the vector have the angle to be measured clockwise from due north

Magnitude (modulus-complex #)

The length of the line represents the magnitude of the vector.

<aside> 💡 Remember compass bearing in the first unit? Below is a reference of both standard and compass bearing.

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Standard Position, Compass Bearing and Degrees, Minutes , Second (DMS)

Property of Vector

• Vectors are parallel if they have the same or opposite direction but not necessarily the same magnitude. In the figure, all of the vectors are parallel except $\vec b$, so all the other vectors are parallel, denoted by: a||c||d||e|
• Vectors are equivalent if they have the same magnitude and direction. In the figure, a = c because they have the same magnitude and direction. Notice that vectors d and e have the same magnitude but opposite directions so they are not equivalent.
• Opposite vectors are those that have the same magnitude, but opposite directions. The vector opposite d is written –d, in the figure –d = e.

<aside> 💡 Equivalent vectors can be parallel, opposite vectors can be parallel. Opposite are not equivalent, and parallel vector didn’t need to be any of them.

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Component Form of a Vector

The component form of a vector (also called wedge form) makes the horizontal and vertical parts of a vector easy to see!