Question

Find the following for the vectors u=3i-10j + √5k and v = - 31 + 10j - √5k.

a. v u, v, and u
b. the cosine of the angle between v and u
c. the scalar component of u in the direction of v
d. the vector proj, u

Answer 1

a.
`\(v \cdot u = -31\)`,
`\(|v| = \sqrt{31^2 + 10^2 + (-√(5))^2} = √(975)\)`,
`\(|u| = \sqrt{3^2 + (-10)^2 + (√(5))^2} = √(134)\)`

b. The cosine of the angle between v and u is
`\((v \cdot u)/(|v| \cdot |u|) = (-31)/(√(975) \cdot √(134))\)`

c. The scalar component of u in the direction of v is
`\((v \cdot u)/(|v|) = (-31)/(√(975))\)`

d. The vector projection of u onto v is
`\(\text{proj}_v u = (v \cdot u)/(|v|^2) \cdot v = (-31)/(975) \cdot (- 31i + 10j - √(5)k)\)`

Explanation:

a. To find
`\(v \cdot u\)`, take the dot product:
`\(v \cdot u = (-31)(3) + (10)(-10) + (-√(5))(√(5)) = -31\)`. For the magnitudes,
`\(|v| = \sqrt{(-31)^2 + 10^2 + (-√(5))^2} = √(975)\)` and
`\(|u| = \sqrt{3^2 + (-10)^2 + (√(5))^2} = √(134)\).`

b. The cosine of the angle between v and u is given by
`\((v \cdot u)/(|v| \cdot |u|) = (-31)/(√(975) \cdot √(134))\).`

c. The scalar component of u in the direction of v is
`\((v \cdot u)/(|v|) = (-31)/(√(975))\).`

d. The vector projection of u onto v is
`\(\text{proj}_v u = (v \cdot u)/(|v|^2) \cdot v = (-31)/(975) \cdot (-31i + 10j - √(5)k)\).`

These calculations involve dot products, magnitudes, and vector projections, providing numerical values for each part based on the given vectors u and v.