Vector Properties Summary Sheet (Accessible HTML)

Vectors

Given Vector A equals the vector with components A sub x, A sub y, and A sub z\(\vec{A}=\langle A_{x},A_{y},A_{z}\rangle\) and Vector B equals the vector with components B sub x, B sub y, and B sub z\(\vec{B}=\langle B_{x},B_{y},B_{z}\rangle\):

• Constant c times Vector A equals the vector with components c times A sub x, c times A sub y, and c times A sub z \(c\vec{A}=\langle cA_{x},cA_{y},cA_{z}\rangle\)
• Vector A plus Vector B equals the vector with components A sub x plus B sub x, A sub y plus B sub y, and A sub z plus B sub z \(\vec{A}+\vec{B}=\langle A_{x}+B_{x},A_{y}+B_{y},A_{z}+B_{z}\rangle\)

Component formulas:

• V sub x equals the magnitude of vector V times the cosine of theta \(V_{x}=||\vec{V}||\cos(\theta)\)
• V sub y equals the magnitude of vector V times the sine of theta \(V_{y}=||\vec{V}||\sin(\theta)\)

Addition of Vectors

For a resultant vector R:

• R sub x equals A sub x plus B sub x \(R_{x}=A_{x}+B_{x}\)
• R sub y equals A sub y plus B sub y \(R_{y} = A_{y} + B_{y}\)
• The magnitude of vector R equals the square root of the quantity R sub x squared plus R sub y squared \(||R||=\sqrt{{R_{x}}^{2}+{R_{y}}^{2}}\)
• Angle phi equals the inverse tangent of the quantity R sub y divided by R sub x \(\varphi=\tan^{-1}(R_{y}/R_{x})\)

Dot Product

Vector A dot Vector B equals the magnitude of vector A times the magnitude of vector B times the cosine of theta \(\vec{A}\cdot\vec{B}=||\vec{A}||||\vec{B}||\cos(\theta)\)

or, given Vector A equals the vector with components A sub x, A sub y, and A sub z\(\vec{A}=\langle A_{x},A_{y},A_{z}\rangle\) and Vector B equals the vector with components B sub x, B sub y, and B sub z\(\vec{B}=\langle B_{x},B_{y},B_{z}\rangle\):

• Vector A dot Vector B equals A sub x times B sub x, plus A sub y times B sub y, plus A sub z times B sub z \(\vec{A}\cdot\vec{B}=A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}\)
• Vector A dot Vector A equals the magnitude of vector A squared \(\vec{A}\cdot\vec{A}=||\vec{A}||^{2}\)

Cross Product

The magnitude of Vector A cross Vector B equals the magnitude of A times the magnitude of B times the sine of theta \(||\vec{A}\times\vec{B}||=||\vec{A}||||\vec{B}||\sin(\theta)\)

• Vector A cross Vector A equals zero \(\vec{A}\times\vec{A}=0\)
• Vector A cross Vector B equals negative open parenthesis Vector B cross Vector A close parenthesis \(\vec{A}\times\vec{B}=-(\vec{B}\times\vec{A})\)

Identities

• Vector A dot the quantity Vector B cross Vector C equals Vector C dot the quantity Vector A cross Vector B \(\vec{A}\cdot(\vec{B}\times\vec{C})=\vec{C}\cdot(\vec{A}\times\vec{B})\)
• Vector A cross the quantity Vector B cross Vector C equals the quantity Vector A dot Vector C times Vector B, minus the quantity Vector A dot Vector B times Vector C \(\vec{A}\times(\vec{B}\times\vec{C})=(\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C}\)