WEBVTT
00:00:00.210 --> 00:00:10.750
Which of the following points lies in the plane three times quantity đť‘Ą plus four minus two times quantity đť‘¦ plus one minus seven times quantity đť‘§ minus six equals zero?
00:00:11.290 --> 00:00:14.090
(A) Three, negative two, negative seven.
00:00:14.580 --> 00:00:17.200
(B) Seven, negative one, negative 13.
00:00:17.620 --> 00:00:20.190
(C) Four, one, negative six.
00:00:20.540 --> 00:00:23.340
(D) Negative four, negative one, six.
00:00:23.730 --> 00:00:33.650
Okay, the way to figure out which of these points lies in the given plane is to substitute in the đť‘Ą-, đť‘¦-, and đť‘§-values given in these points into the plane equation.
00:00:34.030 --> 00:00:40.700
When we do that and calculate the left-hand side of this equation, if it equals zero, then that point does lie in the plane.
00:00:41.000 --> 00:00:47.640
What weâ€™ll do then is substitute in these points into our given plane equation one by one starting with option (A).
00:00:48.140 --> 00:00:57.700
Doing this, we have three times the quantity three plus four minus two times the quantity negative two plus one minus seven times the quantity negative seven minus six.
00:00:58.070 --> 00:01:03.420
This equals three times seven minus two times negative one minus seven times negative 13.
00:01:03.780 --> 00:01:08.520
Thatâ€™s 21 plus two plus 91, or 114.
00:01:08.870 --> 00:01:20.220
Since this result is not equal to zero as the right-hand side of our plane equation has, then we can say that a point with the coordinates three, negative two, negative seven does not lie in this plane.
00:01:20.770 --> 00:01:23.540
Option (A) then is eliminated from consideration.
00:01:23.840 --> 00:01:25.720
Now letâ€™s look at the point in option (B).
00:01:26.030 --> 00:01:35.020
We have three times the quantity seven plus four minus two times the quantity negative one plus one minus seven times the quantity negative 13 minus six.
00:01:35.400 --> 00:01:40.150
Thatâ€™s three times 11 minus two times zero minus seven times negative 19.
00:01:40.610 --> 00:01:46.500
Thatâ€™s 33 plus 133, or 166, also not zero.
00:01:46.880 --> 00:01:52.130
We can see then that this point seven, negative one, negative 13 is also not in the plane.
00:01:52.530 --> 00:02:07.140
Moving on to option (C), three times the quantity four plus four minus two times the quantity one plus one minus seven times the quantity negative six minus six equals three times eight minus two times two minus seven times negative 12.
00:02:07.580 --> 00:02:12.490
This is 24 minus four plus 84, or 104.
00:02:12.860 --> 00:02:17.480
For this point too then, when we plug it into our planes equation, we donâ€™t get zero.
00:02:17.660 --> 00:02:19.480
The point doesnâ€™t lie in the plane.
00:02:19.880 --> 00:02:22.440
Letâ€™s hope we find a different result for option (D).
00:02:22.850 --> 00:02:35.520
Three times the quantity negative four plus four minus two times the quantity negative one plus one minus seven times the quantity six minus six equals three times zero minus two times zero minus seven times zero.
00:02:35.860 --> 00:02:37.360
This all adds up to zero.
00:02:37.550 --> 00:02:43.560
And so we found evidence that this point negative four, negative one, six does lie in the given plane.
00:02:43.990 --> 00:02:45.310
And so thatâ€™s our answer.
00:02:45.490 --> 00:02:56.310
The point negative four, negative one, six lies in the plane three times the quantity đť‘Ą plus four minus two times the quantity đť‘¦ plus one minus seven times the quantity đť‘§ minus six equals zero.