To determine if the vectors are orthogonal, we first calculate their dot product.
The dot product of two vectors is given by:
u . v = (-6)(3) + (4)(3) = -18 + 12 = -6
If the dot product is zero, then the vectors are orthogonal. In this case, the dot product is -6, which means the vectors are not orthogonal.
To find the angle between the two vectors, we can use the formula:
cos(theta) = (u . v) / (||u|| * ||v||)
where theta is the angle between the two vectors and ||u|| and ||v|| are the magnitudes of the vectors.
The magnitude of vector u is:
||u|| = sqrt((-6)^2 + 4^2) = sqrt(36 + 16) = sqrt(52)
The magnitude of vector v is:
||v|| = sqrt(3^2 + 3^2) = sqrt(9 + 9) = sqrt(18)
Now we can plug these values into the formula:
cos(theta) = -6 / (sqrt(52) * sqrt(18)) = -6 / (sqrt(936)) = -6 / 18 = -1/3
Taking the inverse cosine of -1/3, we get:
theta = arccos(-1/3) ≈ 109.47 degrees
Therefore, the angle between the vectors u and v is approximately 109.47 degrees.
given vectors u=(-6,4) and v=(3,3), determine if the vectors are orthogal. if they are not orthogonal, find the angle between the two vectors.
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