Question

For Section 1.3 P4, I understand that the vectors are dependent (since w3 lies on the same plane as the linear combination of w1 and w2). But why are the three vectors in a plane, rather than a line?

Answer 1

This is because w1 and w2 are linearly independent i.e. the only solution to c*w1+d*w2 = 0 is the trivial solution c=0, d=0. Since w1 and w2 are linearly independent, they form a plane in R3 (3 dimensional space over the real numbers). Note that if w1 and w2 were linearly dependent, their linear combinations would only span a line given by e.g. c*w1. And, since w3 and w1 are linearly independent, there would be no combination of w1 and w2 that would produce w3. I hope this helps.

Answer 2

As ovanbc says, w1 and w2 are independent. That means they don't lie on the same line. If they were, one would be a multiple of the other. Two vectors (that aren't collinear) determine a plane, just in the same way that the x- and y- axes determine the x-y coordinate plane. In this case w3 happens to lie on that same plane. Perhaps it's easier to pick three special vectors (in x-y-z 3d space)

(1,0,0)
(0,1,0)
(1,1,0)

The first two are the x-axis and y-axis. All three lie on the plane where z = 0.

Problem 4 is the same concept, except the plane isn't flat on the z axis.It's at a tilt