Vector forms

Refer to Khan academy article.

In Gilbert Strang’s _Intro to Linear Algebra_, it’s different point of view:

For row form or column form:

• Row form: v = (1, 1, -1)
• Column form:

“The reason for the row form (in parentheses) is to save space. But v = (1,1, -1) is not a row vector! It is in actuality a column vector, just temporarily lying down. “

For representing method:

Unit vectors

Simply say, a unit vector is just a vector which length is 1. Kinda like the unit circle idea.
It's also called Engineering notation, or the basis vector.

Which means, it could lie on axis or in between.

What’s it for?
Basically just like the unit circle, make things easier to calculate angles or length or so.
Actually it’s working together with unit circle and all other trigonometric knowledges.
so,
UNIT VECTOR IS RATHER A TRIGONOMETRIC WAY TO DEAL WITH VECTORS.
Easier to think about it, is to think about the Similar graph knowledge in the Dilation section.

Standard Basis Vectors & Unit vector form

It’s also called Unit Basis Vectors.
Unit vector is easy, but unit vector form needs your bit more effort to understand.

THIS FORM DOESN’T PRESENT VECTOR AS A POSITION ANYMORE, RATHER PRESENT IT AS HOW MUCH IT STRETCHES UNIT VECTORS, OR SAY PRESENT IT AS A SCALAR.

Assume that there are TWO unit vectors, one vertical, one horizontal.

How to find a vector's unit vector

More intuitively to solve it just to draw it out and solve it with trigonometry.

Example: Find the unit vector of vector (-8, 5)

Convert between vector forms

HIGHLY RECOMMEND TO REVIEW THE COMPLEX NUMBER CONVERSION SKILLS, HAVING THE VERY SAME IDEA.
Refer to previous note: complex number forms conversion skills.

How to find the direction angle of a vector

It’s the very same problem of the topic Inverse trig function.

NOT EASY! NEED TO CONSIDER LOTS OF CONDITIONS, LIKE QUADRANTS, POSSIBLE SOLUTIONS, PRINCIPLE SOLUTION AND SO. RECOMMEND TO REVIEW INVERSE TRIG FUNCTIONS.

Refer to previous notes: Inverse Trigonometric equations.

Example: Find the direction angle of vector (-2, 5) between 0° to 360°.
Solve:

• tanθ = (-5/2)
• θ₁ = -68.2
• θ₂ = 180° + -68.2 = 111.8 because θ₁ is negative could use this trig identity.
• Vector (-2, 5) is located on Quadrant-2,
• So the answer is 111.8.