12-07: Algebraic Vectors

Torque

________ is also the cross product of two vectors.

scalar multiple

Is a(n) ________ of the original vector (collinear)

right hand rule

The ________ is used to determine the orientation of axes in 3 space.

coordinate system

A 2D ________ has 4 quadrants.

magnitude

The ________ measures the twisting effect of the applied force.

algebraic vector

A(n) ________ is a vector expressed using coordinates as opposed to a magnitude and a direction.

vector projection

Scalar and ________ are more often found using their algebraic vectors.

Torque

________ is measured if the length is given in meters and the force is in Newtons.

coordinate system

A 3D ________ has eight octants.

Collinear vectors

vectors that have the same direction but different magnitude

Finding the perimeter given only the vertices

find each magnitude and add them up

converting to component form

Draw a diagram

Use SOH CAH TOA to find x and y respectively (x and y component)

Put both answers in as coordinates, surrounded by square brackets

The magnitude of an algebraic vector

Found by calculating the vector sum of the x and y components (Pythagorean theorem for hypotenuse)

Unit vectors

vectors that have a magnitude and are written with an ˆon top

î

[1, 0]

ˆj

(0,1)

Adding and subtracting algebraic vectors

To find the sum or difference of 2 or more algebraic vectors, simply add or subtract their components

Multiplying vectors by scalars

When multiplying algebraic vectors by scalars, simply multiply each component by the scalar

Colinear vectors

vectors that have the same direction but different magnitude. They are always scalar multiples of each other

To determine if 2 vectors are collinear, we can determine if there is some scalar k we can multiply both vectors by to make them equivalent

A collinear unit vector to a vector must have the following criteria

Has a magnitude of 1 (units)

Is a scalar multiple of the original vector (collinear)

Finding the perimeter given only the vertices (points)

find each magnitude and add them up

ˆk (unit vector for z)

[0, 0, 1]

Dot product

The dot product always produces a scalar (not a vector) and is sometimes referred to as the scalar product

The dot product formula (above) is often used when we’re given geometric vectors or angles between 2 vectors

We can also determine by: x1x2,y1y2

Cross Product

only for 3D vectors
Subtract what you have crossed filling out the x, y and z component