TAM 2xx References

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Vectors and Scalars

Scalars vs. Vectors

Vectors

From TAM212 Reference Pages (Vectors and Bases):

A vector is an arrow with a length and a direction. Just like positions, vectors exist before we measure or describe them. Unlike positions, vectors can mean many different things, such as position vectors, velocities, etc. Vectors are not anchored to particular positions in space, so we can slide a vector around and locate it at any position.
Fig: VectorDef

Taken from TAM212 Reference Pages (Vectors and Bases)

Scalars

While a vector represents magnitude and direction, a scalar is a number that represents a magnitude, but with no directional information. Some examples of scalar quantities can be mass, length, time, speed, or temperature.

Vector Operations

Scaling Vectors

Taken from TAM212 Reference Pages (Vectors and Bases)

Vectors can be multiplied by a scalar number, which multiplies their length.
Fig: VecScaling

Taken from TAM212 Reference Pages (Vectors and Bases). Don't include the a+b vector - only keep one a and one b vector with their scaled counterparts.

Vector Addition and Subtraction

From TAM212 Reference Pages (Vectors and Bases):

Vectors can be added or subtracted together, using the parallelogram law of addition or the head-to-tail rule.
Fig: VecAddition

Taken from TAM212 Reference Pages (Vectors and Bases). Only include the left part of the figure, and add another part on the right that shows a-b.

Unit Vectors

Use the same content here as from the TAM212 Reference Pages (Vectors and Bases - Unit Vectors)

Vector Magnitude and Direction

Vectors can be written as a magnitude (length) multiplied by the unit vector in the same direction as the original vector.
$$ \vec{A} = \|\vec{A}\| \hat{u_A}\ $$

Insert the same content here as from the TAM212 Reference Pages (Vectors and Bases - Length of Vectors) for vector magnitude.

The direction of a vector can be written as a unit vector by dividing the vector components by the vector magnitude.
Fig: UnitVec

Taken from TAM210 Lecture Notes - Slide 3

Alternatively, the vector components can be determined geometrically via the angles of each component with respect to the Cartesian coordinates.

Can probably make some sort of interactive figure here based on the image from TAM 210 lecture notes - slide 4

Fig: UnitVecAngles

Taken from TAM210 Lecture Notes - Slide 4

Scalar and Vector Products

Dot Product

Insert the same content here as from the TAM212 Reference Pages (Vectors and Bases - Dot Product).

Cross Product

Add something here about the three dimensional cross product?

Insert the same content here as from the TAM212 Reference Pages (Vectors and Bases - Cross Product).

Vector Projection

Insert the same content here as from the TAM212 Reference Pages (Vectors and Bases - Projection and Complementary Projection).