Finding the angle between two vectors. In the exercise find the angle theta between the vectors. We have u is a vector negative 4, 3. v is the vector 0, 5 and the inner product of u and v is 3 times u1v1 plus u2v2. So we wanna find the angle theta between the two vectors which means that we can use the formula cosine theta is equal to the inner product of u and v divided by the magnitude of u times the magnitude of v, were for this form we need to use this inner product that’s define rather than the standard dot product. So this will be equal to for the numerator will have 3 times u1 which is negative 4 times v1 which is 0 plus u2 which is 3 times v2 which is 5 this will be divided by the magnitude of u is going to be the squared root of 3 times u1 just negative 4 squared plus u2 which is 3 squared and this will be multiplied by the squared root of 3 times v1 which is 0 squared plus v2 which is 5 squared This will be equal to 15 divided by the squared root of 57 for the first quadratic times 5 for the second. We can divide out the 5 between the numarator and the denominator so we will be left with 3 divided by the squared root of 57 and this is what will be equal to the cosine of theta taking the inverse cosine of both sides will have the theta is equal to cos inverse of 3 divided by squared root of 57 which is approximately equal to 1.16 radiance or 66.59 degrees.

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