How do I write an equation for a 3d line here:

Question

499.Let A = (7,1,1) and B = (−3,2,7). Find all the points P on the z-axis that make

angle APB right.

Because when the two planes intersect, there's a line that goes three-dimensional. How do I write that equation?

7x+y+z=0 and -3x+2y+7z=0

Answer 1

P is (0,0,z) z any real number

AP is effectively vector (7,1,1-z)

PB is effectively vector (3,-2,z-7)

if scalar vector product is zero, vectors are perpendicular (orthogonal)

want

0=21-2-z²-7+8z

solving for z,,,z=4+/-√28

P=(0,0,4+√28)

or

P=(0,0,4-√28)

Answer 2

Have you tried using the Pythagorean Theorem?

If you construct triangle APB in the 3-D plane AND you want to make sure angle APB is right, then:

(AP)²+(BP)² = (AB)²

of course, point P will be (0, 0, z), since it is on the z-axis.

Use the distance formula to find the length of each side, using z as the unknown.

For example: (AP)² = (7-0)² + (1-0)²+ (1 - z)²

Do the same to find AB² and AP²

From there you can set up an equation to solve for the values of z and have your two points that make the angle right.