Scalar and vector projections using dot products.
Example 1
Find the scalar projection of u=(4,6) onto v=(1,0).
Drop a perpendicular from u to the line in the direction of v. Both axes use the same scale.Show worked answer
Answer: 4
- Scalar projection =∣v∣(u⋅v).
- The dot product is 4 and ∣v∣=1, so the scalar projection is 4.
Example 2
Find the vector projection of u=(2,7) onto v=(0,3).
Drop a perpendicular from u to the line in the direction of v. Both axes use the same scale.Show worked answer
Answer: (0,7)
- Vector projection =(v⋅vu⋅v)v.
- The coefficient is 921; multiplying by (0,3) gives (0,7).
Example 3
Find the scalar projection of u=(9,2) onto v=(1,0).
Drop a perpendicular from u to the line in the direction of v. Both axes use the same scale.Show worked answer
Answer: 9
- Scalar projection =∣v∣(u⋅v).
- The dot product is 9 and ∣v∣=1, so the scalar projection is 9.