Trigonometry

Question

Find the angle between the given vectors

u=-i-4j, v=3i-3j

A. -0.9

B. 59

C. 149

D. 74.4

Answer 1

Hello Leonel.

I'll pick (B)  59  degrees.

The formula is cos(X) =   (U . V)/( |U||V| )

The english translation of this formula is the cosine of the angle in between the

vectors U and V is equal to their dot product divided by the product of their lengths.

 

The length of the vector U = square root (  (-1)^2 + (-4)^2 ) ) 

                                      = square root (    1      +   16   )

                                      = square root ( 17)

 

The length of the vector V = square root ( (3)^2 + (-3)^2 )

                                      = square root ( 9 + 9)

                                      = square root (18)

                                      = square root (9 *2)

                                      = 3 * square root(2)

 

The dot product of these vectors is ( -1 * 3) + (-4 * -3) = -3 + 12 = 9

 

So the right side of the equation is 9/(square root(17)*3*square root(2))

                                 = 9 / (3 * square root(34))

 

The 9 and 3 cancel and the denominator must be rationalized.

 

The final expression for the right side is  3*square root(34)/34 =  0.5144957554275265

 

The cosine of the angle is equal to this value , correct to 15 digits. Recall that the cosine of 60 is 1/2 = 0.5.

 

Taking the inverse cosine of this decimal value or as it is sometimes called arc-cos

 

gives the answer of just over 59 degrees.