Homework: Vectors (Section 8.4)

Question

A person leaves home and walks 3 miles west, then 4 miles southwest.

How far from home is she?

   miles:

In what direction must she walk to head directly home?

   degrees North of East:

Answer 1

the distance from home is the hypotenuse of a right triangle with sides 2sqr2 and 3 + 2sqr2

d^2 = 8 + 9 + 8 + 12sqr2 = 25 + 12sqr2 = about 25+16.8 = 41.8

d = sqr41.8 = 6.5 from home

her angle back home is NE but substantially less than 45 degrees

angle is the sin^-1(2sqr2/(3+2sqr2)= sin^-1(2.8/5.8) = sin^-1(.483) = 28.9 degrees

Use a calculator.

Answer 2

For the first problem, you can either use vectors, squaring, and square roots, or the cos law.

The cos law gives you √3^2+4^2-2*3*4*cos(135°) = √(25+12√2)

With vectors, you would get (-3, 0) + (-2√2, -2√2) = (-3-2√2),-2√2), which is a distance

√((-3-2√2)^2+(-2√2)^2) = √(9+12√2+8+8) = √(25+12√2), the same result. This is approximately 6.4785.

As the first line is due west, the angle between this line and the hypotenuse is what we want, as we are looking at the angle going back in the opposite direction, which will be the line going the opposite direction versus the line going east.

Using the sin law, as this angle is opposite the side of length 4, we get sin^-1(4 * sin(135°)/√(25+12√2)) =

sin^-1(2√2/√(25+12√2)) =25.89°

25.89° N of E.