ObsidianVault/Wiki/Math/Vector Basics.md

#rs/class/math163 #math

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Operations

Addition

\vec{u} + \vec{v} = <u_{1} + v_{1}, u_{2} + v_{2}>

Subtraction

\vec{u} - \vec{v} = <u_{1} - v_{1}, u_{2} - v_{2}>

Dot Product

\vec{u} \cdot \vec{v} = <u_{1}v_{1}, u_{2}v_{2}>

Magnitude

||\vec{u}|| = \sqrt{ u_{1}^2 + u_{2}^2}

Unit Vector

unit_u=\frac{\vec{u}}{||\vec{u}||}

Used to find the vector of magnitude 1 in the same direction as u

Cross Product

a \times b = \begin{vmatrix} i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}= i(a_2b_3-a_3b_2)-j(a_1b_3-a_3b_1)+k(a_1b_2-a_2b_1)

Used to create a vector that is orthogonal to both a and b. To find what direction it will point in use the right-hand rule (thumb, pointer and middle finger).

Properties:

• Not commutative u\times v \ne v \times u
• Anti-commutative u \times v = -(v \times u)
• On self a \times a = 0 The magnitude of the cross product of two vectors can be found with ||u \times v|| = ||u|| * ||v|| * \sin(\theta) The magnitude of the cross product is equal to the area of the parallelogram formed with two adjacent sides as the vectors.

Projection

proj_{u}v=\frac{u\cdot v}{\left|\left|u\right|\right|^{2}}u

Used to project vector v onto u !VectorProjection.excalidraw

Triple Scalar Product

Find the Determinant of:

u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \end{vmatrix}

The triple scalar product can also be found by finding the cross product of v and w and then taking the dot product of the resulting vector and u.

u * (v \times w) = (u \times v) * w

The triple scalar product is equal to the volume of the parallelpiped where each vector represents one adjacent edge.