MATH SOLVE

8 months ago

Q:
# Let vec u =<3,-1> and vec v =<1; 2 > Find , compu and proj -> u
step by step pls

Accepted Solution

A:

Sure, let's find the components, unit vector, and projection of vector u step by step.
Step 1: Find the magnitude of vector u (||u||).
The magnitude of vector u (||u||) is given by the formula:
||u|| = β(uβΒ² + uβΒ²)
Given vector u = <3, -1>, the magnitude of u is:
||u|| = β(3Β² + (-1)Β²) = β(9 + 1) = β10 β 3.162
Step 2: Find the unit vector of u (u_hat).
The unit vector of u (u_hat) is the vector in the same direction as u but with a magnitude of 1. It is obtained by dividing each component of u by its magnitude.
u_hat = u / ||u||
Given vector u = <3, -1>, and ||u|| = β10 β 3.162, the unit vector u_hat is:
u_hat = <3/β10, -1/β10> β <0.949, -0.316>
Step 3: Find the components of vector u (u_parallel and u_perpendicular).
To find the components of vector u, we need a reference vector v that is parallel to u. We'll use vector v = <1, 2> for this purpose.
The parallel component of u (u_parallel) with respect to v is given by the formula:
u_parallel = (u Β· v_hat) * v_hat
where v_hat is the unit vector of v.
First, find v_hat (the unit vector of v):
v_hat = v / ||v||
Given vector v = <1, 2> and ||v|| = β(1Β² + 2Β²) = β5 β 2.236
v_hat = <1/β5, 2/β5> β <0.447, 0.894>
Now, find u_parallel:
u_parallel = (u Β· v_hat) * v_hat
u_parallel = (<3, -1> Β· <0.447, 0.894>) * <0.447, 0.894>
u_parallel = (1.341 - 0.894) * <0.447, 0.894>
u_parallel = 0.447 * <0.447, 0.894>
u_parallel = <0.2, 0.4>
The perpendicular component of u (u_perpendicular) with respect to v is given by the formula:
u_perpendicular = u - u_parallel
u_perpendicular = <3, -1> - <0.2, 0.4>
u_perpendicular = <3 - 0.2, -1 - 0.4>
u_perpendicular = <2.8, -1.4>
Step 4: Find the projection of u onto v (proj_u_v).
The projection of u onto v (proj_u_v) is the vector that represents the component of u in the direction of v. It is equal to u_parallel.
proj_u_v = u_parallel = <0.2, 0.4>
Summary:
Magnitude of u (||u||) β 3.162
Unit vector of u (u_hat) β <0.949, -0.316>
Parallel component of u (u_parallel) β <0.2, 0.4>
Perpendicular component of u (u_perpendicular) β <2.8, -1.4>
Projection of u onto v (proj_u_v) β <0.2, 0.4>