Find the angle between 𝒖 = −2, 5 , and 𝒗 = 1, −3

To find the angle between two vectors, we can use the formula:

cos θ = (u.v) / (||u|| ||v||)

where:

• u and v are the vectors,
• "u.v" is the dot product of u and v,
• "||u||" is the magnitude of vector u,
• "||v||" is the magnitude of vector v,
• "θ" is the angle between the vectors.

Step 1: Calculate the dot product (u.v) u.v = (-21) + (5-3) = -2 - 15 = -17

Step 2: Calculate the magnitude of u (||u||) ||u|| = sqrt((-2)^2 + 5^2) = sqrt(4 + 25) = sqrt(29)

Step 3: Calculate the magnitude of v (||v||) ||v|| = sqrt((1)^2 + (-3)^2) = sqrt(1 + 9) = sqrt(10)

Step 4: Substitute these values into the formula cos θ = -17 / (sqrt(29) * sqrt(10))

Step 5: Solve for θ θ = cos^-1(-17 / (sqrt(29) * sqrt(10)))

This will give you the angle between the two vectors.