Vector Operations
Vectors can be manipulated mathematically to describe physical quantities and their interactions. In this section we introduce the most common vector operations used in statics: scalar multiplication, vector addition, and vector subtraction.
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation changes the magnitude of the vector while maintaining its direction. If the scalar is negative, the direction is reversed.
Multiply by 2
Doubles the vector magnitude.
Multiply by −1
Reverses the vector direction.
Multiply by 1/2
Halves the vector magnitude.
Vector Addition
Vector addition combines two or more vectors to produce a resultant vector. Graphically, vectors are added using the tip-to-tail method.
Tip-to-tail method
Place the tail of one vector at the tip of the other, then draw the resultant from the tail of the first vector to the tip of the last.
Triangle method and commutativity
The triangle method demonstrates that vector addition is commutative: \(\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}\).
Vector Subtraction
Vector subtraction can be interpreted as adding a vector in the opposite direction. That is: \[ \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B}) \]
Graphical interpretation
Draw \(-\mathbf{B}\) by reversing the direction of \(\mathbf{B}\), then add using the tip-to-tail method.
