given factors u=-6,4 and v=3,3, determine if the vectors are orthogonal. if they are not orthogonal, find the angle between the two vectors.
Options:
- The vectors are orthogonal
- The vectors are not orthogonal. The angle between the two vectors is 72.5°.
- The vectors are not orthogonal. The angle between the two vectors is 101.3°.
- The vectors are not orthogonal. The angle between the two vectors is 130.6°.
To determine if the vectors are orthogonal, we need to calculate the dot product of the two vectors:
u ∙ v = (-6)(3) + (4)(3) = -18 + 12 = -6
If the dot product of two vectors is 0, then the vectors are orthogonal. Since the dot product of u and v is not 0, the vectors are not orthogonal.
To find the angle between the two vectors, we can use the formula:
cosθ = (u ∙ v) / (||u|| * ||v||)
Where ||u|| and ||v|| are the magnitudes of the vectors.
||u|| = √((-6)^2 + 4^2) = √(36 + 16) = √52
||v|| = √(3^2 + 3^2) = √(9 + 9) = √18
Plugging in the values:
cosθ = (-6) / (√52 * √18)
cosθ = -6 / (√(52 * 18))
cosθ = -6 / 6√13
cosθ = -1 / √13
cosθ = -√13 / 13
Therefore, the angle between the two vectors is arccos(-√13/13) ≈ 101.3°.
So, the correct answer is:
- The vectors are not orthogonal. The angle between the two vectors is 101.3°.