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Wize University Dynamics Textbook (Master) > Transition to Dynamics

Review of Vector

Review of Useful Trigonometric Equations & Laws of Sin & Cos Previous Section
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  • Scalar: a quantity with magnitude but not direction
  • Vector: a quantity with both magnitude and direction
  • Unit vector: a vector with magnitude of 1, typically used to indicate direction
  • Directional angles: the angles between a vector and the positive axis
Properties of Vectors

Magnitude of a vector
A= Ax2+Ay 2 +Az2A=\ \sqrt{A_x^{2_{ }}+A_{y\ }^{2\ }+}A_z^2

Converting into unit vectors

uA⃗ =A⃗ A=AxAi +AyAj +AzAk\vec{u_A}^{\,}=\frac{\vec{A}^{\,}}{A}=\frac{A_x}{A}i\ +\frac{A_y}{A}j\ +\frac{A_z}{A}k

Dot product

A⃗ ⋅B⃗ =ABcos⁡θ\vec{A}^{\,}\cdot \vec{B}^{\,}=AB\cos\theta
A⃗ ⋅B⃗ =AxBx+AyBy+AzBz\vec{A}^{\,}\cdot \vec{B}^{\,}=A_xB_x+A_yB_y+A_zB_z
θ=cos⁡−1(A⃗ ⋅B⃗ AB)\theta=\cos^{-1}\left(\frac{\vec{A}^{\,}\cdot \vec{B}^{\,}}{AB}\right)

Can also be thought of as a measure of how parallel two vectors are

Properties of Directional Angles

Unit vector with directional angles
uA⃗ =cos⁡αi+cos⁡βj+cos⁡γk\vec{u_A}^{\,}=\cos\alpha i+\cos\beta j+\cos\gamma k
Relationship between directional angles

cos⁡2α+cos⁡2β+cos⁡2γ=1\cos^2\alpha^{ }+\cos^2\beta+\cos^2\gamma=1


Vector Manipulation

Things you should know:
  • How to break down a vector into perpendicular components
  • How to break down a vector into non-perpendicular component
  • Projecting a vector along a line
  • Finding the angle between two vectors
Q: How to break down a vector into perpendicular components?
A: Basic Trigonometric rules, aka. SOH CAH TOA

Q: How to break down a vector into non-perpendicular components?
A: Sine and cosine laws

Q: How to project a vector along a line
A: Using the dot product

Q: Finding the angle between two vectors
A: Modified dot product