Definitions

Co-linear: Same line Co-planer: Same plane

Every vector coplanar with a and b can be expressed as a linear combination of the two.

Vectors are equal if their midpoints are the same.

The vector $\overrightarrow{AB}$ has starting point A and ending point, terminal point, B.

Position vector: starting at origin and ending on point P.

Magnitude is just distance from the start and ending point, described with Pythagorean, and the notation $|\vec{A}|$.

Unit vector can be obtained by dividing the vector by its magnitude.

Dot Product

Algebraic definition: $\mathbf{u}\cdot \mathbf{v}=u_1{v}_1+u_2{v}_2+\cdots +u_n{v}_n$ Geometric definition: $\mathbf{u}\cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos \theta$ If the dot product is zero, the two vectors are perpendicular. We can rearrnage, to show that the dot product also gives the projection of one vector onto the other.

The projection of $\mathbf{u}$ on $\mathbf{v}$ is given by $|\mathbf{u}|\cos \theta$. Hence, $\frac{\mathbf{u}\cdot \mathbf{v}}{|v|}$ gives the projection.//

If they are parallel, $a\cdot b=\pm |a||b|$. The negative case is when the vectors are going in opposite directions, since $\cos \pi =-1$. Also, $a\cdot a=|a{|}^2$

Equation of Lines

Vector equation:

Parametric Form:

Cartesian Form

Understand where the directional vector and positional vector are in each form. They are all there.

A pair of lines can have three states:

• Intersection
• Parallel
• Not intersect nor parallel (skewed)

Cross product

$\mathbf{a}\times \mathbf{b}=(a_2{b}_3-a_3{b}_2)\mathbf{i}-(a_1{b}_3-a_3{b}_1)\mathbf{j}+(a_1{b}_2-a_2{b}_1)\mathbf{k}$ An important property: $\mathbf{a}\times \mathbf{b}=(|a||b|\sin \theta )\hat{n}$ where $\hat{n}$ is the unit vector whose direction is given by the right hand rule to $a$ and then $b$.

So, if we only take the magnitude of both sides, we will obtain the other identity: $|\mathbf{a}\times \mathbf{b}|=|a||b|\sin \theta$ Examining the right side, we can also see that this magnitude is the same as the area of the parallelogram enclosed by the two vectors $a$ and $b$.

Operation properties:

• Non-communitive
• Non-associative
• Distributive
• You can distribute a scalar into ONE of the vectors. ie. $\lambda (a\times b)=\lambda a\times b=a\times \lambda b$

Mixed Product

The mixed product $|(a\times b)\cdot c|$ gives the volume of the parallelepiped formed by three non-coplaner vectors. The proof is as follows: