If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?
Practice Questions
1 question
Q1
If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?
(1, 2, 3)
(2, 4, 6)
(0, 0, 0)
(x, y, z)
Projection of P onto a = ((P · a) / |a|^2) * a.
Questions & Step-by-step Solutions
1 item
Q
Q: If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?
Solution: Projection of P onto a = ((P · a) / |a|^2) * a.
Steps: 6
Step 1: Identify the position vector of point P, which is given as (x, y, z).
Step 2: Identify the vector a, which is given as (1, 2, 3).
Step 3: Calculate the dot product of P and a, denoted as P · a. This is done by multiplying the corresponding components of the vectors and adding them together: (x * 1) + (y * 2) + (z * 3).
Step 4: Calculate the magnitude squared of vector a, denoted as |a|^2. This is done by squaring each component of a and adding them together: (1^2) + (2^2) + (3^2) = 1 + 4 + 9 = 14.
Step 5: Divide the result from Step 3 (P · a) by the result from Step 4 (|a|^2) to get the scalar value for the projection: ((P · a) / |a|^2).
Step 6: Multiply the scalar value from Step 5 by the vector a to get the projection of P onto a: ((P · a) / |a|^2) * a.
