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)