Let u= <3,1> and v=<6,-1>
A. what is the angle between u and v?
B. find projvU
C. reoslve u into u1 parallel to v abd U2 perpendicular to v.
u dot v = |u| |v| cosØ where Ø is the angle between them
18-1 = √10√37cosØ
cosØ = 17/√370
Ø = cos^-1 (17/√370) = appr 27.9°
your B and C are undecipherable
B. find projected vU
C. resolve u into u1 parallel to v and U2 perpendicular to v.
To answer these questions, we will use the properties and formulas related to vector operations. Let's go step by step:
A) To find the angle between two vectors, we can use the dot product formula and the relationship between dot product and cosine of angles:
The dot product formula: u · v = |u| |v| cosθ, where θ is the angle between the vectors.
So, let's calculate the dot product of u and v using their components:
u · v = (3)(6) + (1)(-1) = 18 - 1 = 17
Now, let's find the magnitudes of the vectors:
|u| = √(3^2 + 1^2) = √10
|v| = √(6^2 + (-1)^2) = √37
And now we can solve for the angle θ:
17 = (√10)(√37) cosθ
cosθ = 17 / (√10)(√37)
θ = arccos(17 / (√10)(√37))
Using a calculator, we find that θ ≈ 11.6 degrees.
B) To find the projection of u onto v (projvU), we can use the formula:
projvU = (u · v / |v|²) v
Let's plug in the values we already know:
projvU = (17 / 37)(6, -1)
projvU = (6(17/37), -1(17/37))
projvU ≈ (2.541, -0.459)
C) To resolve u into u1 parallel to v and u2 perpendicular to v, we can use the projection and subtraction formulas:
u1 = projvU
u2 = u - u1
Let's calculate:
u1 = (2.541, -0.459)
u2 = (3, 1) - (2.541, -0.459)
u2 = (0.459, 1.459)
Therefore, we have:
u1 = (2.541, -0.459)
u2 = (0.459, 1.459)