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I would like to verify this problem from an introductory to Linear Algebra course.

It goes as follows:

From the given parametric equations I constructed the vectors:

line L:

To find w1, I know that

And to find k: (v.L)/||L||

And

The issue I am facing is, which vector do I chose for the L?

I have found 2 vectors from the parametric equations.

Should I simply take the difference?

Thank You.

[edit.]

If I stick to my L line being equal to (1,-3,3), due to the fact that b is my position vector, then:

Am I completely off?

It goes as follows:

This is how I proceeded:Let L be the line with parametric equations x=2+3t, y=1-2t, z=-2+t, and letv=(3,2,2). Find vectorsw1 andw2 such thatv=w1+w2, and such thatw1 is parallel to L andw2 is perpendicular to L.

From the given parametric equations I constructed the vectors:

line L:

**a**=(3, -2, 1) and**b**=(2,1,-2).To find w1, I know that

**w**1=*k*LAnd to find k: (v.L)/||L||

^{2}And

**w**2 is just a matter of:**w**2=v-**w**1The issue I am facing is, which vector do I chose for the L?

I have found 2 vectors from the parametric equations.

Should I simply take the difference?

**a**-**b**= (1,-3,3)?Thank You.

[edit.]

If I stick to my L line being equal to (1,-3,3), due to the fact that b is my position vector, then:

**w**1 = 3/19(1,-3,3)**w**2 = (3,2,2) - 3/19(1,-3,3)Am I completely off?

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