# MA 227 Standard S02

The Dot Product

At the end of the course, each student should be able to…

• S02: DotProd. Compute and apply the dot product of two vectors.

## S02. The Dot Product

• The dot product $$\vect v\cdot\vect w$$ measures the work done by a force given by $$\vect v$$ along a displacement given by $$\vect w$$.
• If the force is in the same direction of the displacement, then work is simply given by scalar multiplication of the magnitudes of $$\vect v,\vect w$$.
• If the force is orthogonal to displacement, then no work is done.
• It may be computed by $$\vect v\cdot\vect w=\|\vect v\|\|\vect w\|\cos\theta$$ where $$\theta$$ is the angle between the vectors.
• The following properties may be proven about the dot product.
• $$\vect{v} \cdot \vect{w} = \vect{w}\cdot\vect{v}$$
• $$(c\vect{v})\cdot \vect{w} = \vect{v} \cdot (c\vect{w}) = c(\vect{v} \cdot \vect{w})$$
• $$\vect{v} \cdot (\vect{w}_1 + \vect{w}_2) = \vect{v}\cdot\vect{w}_1 + \vect{v}\cdot \vect{w}_2$$
• $$\vect{v} \cdot \vect{v} = \|\vect{v}\|^2$$
• $$\vect{0} \cdot \vect{v} = 0$$
• These properties may be used to obtain a simplified formula: $$\vect v\cdot\vect w=v_xw_x+v_yw_y+v_zw_z$$.
• This allows $$\theta$$ be easily computed from $$\cos\theta=\frac{\vect v\cdot\vect w}{\|\vect v\|\|\vect w\|}$$.
• Thus two nonzero vectors are orthogonal if and only if $$\vect v\cdot\vect w=0$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 11.3 (exercises 1-8[parts a,b], 31-32, 41)