Tutorial Exercise Determine whether the planes a1x + b1y +...

Tutorial Exercise Determine whether the planes $a_1x + b_1y + c_1z = d_1$ and $a_2x + b_2y + c_2z = d_2$ are parallel, perpendicular, or neither. The $a_1a_2 + b_1b_2 + c_1c_2 = 0$. $6x + y - 5z = 4$, $-42x - 7y + 35z = 5$ Step 1 Compare the given equations with the equations $a_1x + b_1y + c_1z = d_1$ and $a_2x + b_2y + c_2z = d_2$ then $a_1 = 6$, $b_1 = 1$, $c_1 = -5$, $a_2 = -42$, $b_2 = -7$, and $c_2 = 35$ Step 2 Observe the following. $a_2 = -42$ $= (-7)($oxed{6})$ $= oxed{ }$ $a_1$

Transcript

00:01 Okay, so part a of this problem asks us to rewrite the system of equations here as a matrix equation, ax equals b.

00:08 And so to do this, i always just write out the outline of what this equation looks like.

00:14 So this will be your a on the far left, multiply by a skinny x vector, and that's going to equal a skinny b vector.

00:24 And then i'll write in my variables here like this, xyz.

00:29 It's important that the order stays the same.

00:31 So i always write x, y, and z above my column, so i know which coefficients are going where.

00:37 And this b vector is going to be your constants, and it's important that those constants come after the equal sign.

00:43 So make sure that they do in your original equation.

00:46 No issues here, but in other problems, it could be an issue.

00:49 And then you go line by line.

00:51 So the first line, we're just going to write it like this.

00:54 So it's 1x minus 6 y's plus 3 z's, equal.

00:58 11.

01:01 Then the next row would be 2 x's minus 7y.

01:05 You should all be ys up here, plus 3z equals 14.

01:14 And then the last row would be 4x minus 12y plus 5 z equals 25.

01:21 And so this would be the matrix equation that you would want to set up that you'd be trying to solve to get these variables right here, xx, y, so part b of this problem, asks us to take this matrix equation and solve it using a inverse here.

01:37 A is given by the matrix that we solve for in part a, which looks like this.

01:43 B is this skinny vector over here.

01:45 And the way that we're going to do this is we're going to say, okay, you want to solve ax equals b.

01:51 And to do that, we're going to multiply a inverse by both sides.

01:55 When we do that, a inverse is going to cancel out with a over here and just leave us with x, which is what we're trying to solve for, and all we need to do is multiply a inverse and b.

02:06 So let's do that here.

02:08 We'll take a inverse, which is interestingly the same thing as a above, and we're going to multiply that by our b, and this should give us our answer...

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