course mth277

In reference back to this problem, which involves two distinct unit vectors orthogonal to v=i+j-k and w=-i+j+kso we have

x=a1i+a2j+a3k

v=i+j-k

w=-i+j+k

I did get a little confused on doing this, but hopefully you can clarify (we may have begun a discussion about some other topic)

Here's what I have written down:

the magnitude of the vector x= 1, so 1=a1+a2+a3

a1^2 + a2^2 + a3^2 = 1. which is not the same thing as you've written

x(dot)v=0

therefore x=a1(1)+a2(1)+-1(a3)= 0

(not sure if this is right...how can this equal zero)

But anyways, we have three equations of unknowns(as you said, but what is the third equation):

we have vector-x (dot) vector-w= -a1+a2+a3=0

then vector-x (dot) vector-v= a1+a2-a3=0

It asks for two distinict orthogonal vectors , so do I need another vector (call it vector y) for which it is orthogonal to both vector v and vector w?

(I also briefly reviewed solving an equation with three unknowns; I remembered doing it in 173 with the quadratic model of depth vs. clock time.

How would we use vector methods to prove that an angle inscribed in a semicircle must be a right angle?

(key word prove, which Im sure involves the geometric proof that I've seen in the chapter notes)."

You have all the required equations:

-a1+a2+a3=0
a1+a2-a3=0
a1^2 + a2^2 + a3^2 = 1.

This is a system of simultaneous equations in three unknowns a1, a2 and a3.

You solve by elimination, substitution or a combination of the three.

For example, solve the first equation for a1, then substitute the resulting expression for a1 in the remaining two equations. You will obtain two equations in two unknowns.

Then solve the first of these two new equations for a2 in terms of a3. Substitute this expression for a2 into the second of your new equations. You get a quadratic equation in the single unknown a3. Solve it (you get two so lutions).

Then back-substitute to find a2 and a1.