00:00Hi guys, so in this
00:02next lesson we are going
00:03to find the angle between
00:05a line and a plane.
00:06So I think of a
00:06line coming straight down and
00:09going I don't know through
00:10a plane like this. It's
00:11this angle that we're trying
00:13to find here. It might
00:15necessarily be right here, but
00:18I think of it. I
00:19like to think of it.
00:20Get a piece of string
00:22coming down and touching touching
00:24the table at an angle.
00:25That's the angle we're looking
00:26for. Or this example
00:28I was a ray of
00:28light coming from a point
00:30coming down and maybe hitting
00:31the ground. On geodebra, it
00:34looks something like this. So
00:35here's our plane. Now, if
00:39you zoom in like when
00:41we're, where I first had
00:42it here, it doesn't even
00:43touch the plane, but a
00:45line will always, and we
00:46talked about this when we
00:47got the intersecting point, but
00:48a line's always going to
00:49touch a plane. It's always
00:50going to intersect a plane
00:52unless, unless they're
00:56parallel. So when they talk
00:57about when he asks for
00:58the angle between the line
01:00and the plane, he's looking
01:00at this angle here. And
01:05of course, there's this angle
01:06too. So there's an acute
01:07angle and an obtuse angle.
01:08So just be careful what's
01:12he asking for. OK, so
01:14let's do an example. Well,
01:17actually, before I do this,
01:18let me show you how
01:19we do it. So how
01:21am I supposed to get
01:22the angle between
01:24line and a plane if
01:25I like I have the
01:26plane but I don't have
01:28I don't have a vector
01:29here I don't have this
01:31vector I don't have any
01:32vector on the line but
01:34hopefully you've started to see
01:36now that when it when
01:37it comes to planes the
01:38most important the most important
01:41part of the plane or
01:43part of the equation of
01:44the plane is the normal
01:45because the normal kind of
01:46gives us it's actually perpendicular
01:48to the direction but it's
01:51it's our way of kind
01:52of looking
01:52at directions because it's very
01:53easy to see when it's
01:56in this form. So the
01:58normal is the normal comes
02:02straight out of the plane
02:03like this. It goes straight
02:05up now you have to
02:05imagine this is 3D, this
02:07is a table and the
02:08normal is coming straight out
02:10of the table. This is
02:11called n for normal. Well
02:15the angle between
02:20the angle between the line
02:22and the plane, let's call
02:23theta. Now again, as I
02:25say, I don't know what
02:26this, I don't have a
02:27vector on the plane. So
02:29I just don't know what
02:29this vector is. I can't
02:31use it to find the,
02:33um, this angle. But I
02:36can find this angle and
02:38we often call this theta
02:40and this angle phi. So
02:42this angle is phi. I
02:44can find phi because phi
02:45is the angle between the
02:47normal and
02:48the line. And I know
02:49how to find the the
02:50angle between a vector and
02:52a line. I just use
02:53the direction vector of the
02:55line. So I can find
02:57phi. And then theta, once
02:59I have phi, I can
03:01just get theta because theta
03:02is just, I don't know
03:04if we're, often we use
03:05degrees for vectors, but so
03:07theta would just be 90
03:08degrees minus phi. That's obvious
03:12because this is a right
03:12angle. And then I'm not
03:14for it 90. So if
03:16this is
03:16If I have 5, this
03:18is 30, then this is
03:20obviously 60. And that's it.
03:23So to get the angle
03:25between a line and a
03:26plane, we get the angle
03:27between the normal and the
03:30line, and then subtract that
03:32from 90 degrees. If you
03:34end up getting 5 and
03:35you end up getting an
03:35obtuse angle, make sure you're
03:37using the acute angle because
03:38of this diagram that I've
03:39just drawn. Okay, let's do
03:41this example. A ray of
03:42light coming from the point,
03:44is traveling in the direction
03:45of this. So our line,
03:47our line, let's call it
03:49L, is R equals point
03:53or position vector negative one,
03:56three, two. Let me answer
03:59my pen. So the light,
04:01the position vector is negative
04:04one, three, two. And the
04:05direction vector, it gives us
04:06here, is four negative one,
04:09two. So let me write
04:09that again. Negative one, three,
04:12two,
04:12plus direction vector t times
04:154 1 negative 2. And
04:19then the plane is given
04:26as this. Now, as I
04:28always say, when you can
04:31with vectors draw a diagram.
04:33So let's say, and again,
04:34it doesn't have to be
04:35flat. Let's say this is
04:36the plane. Remember, you're kind
04:38of looking at it from
04:40the side. So this is
04:41the plane pi and this
04:43is my straight line, my
04:46straight line coming like this.
04:48You can draw in any
04:50way you want. This is
04:52my line L and then
04:54the normal comes straight out,
04:59straight out of the plane
05:01like this normal. Now my
05:05normal, so the normal N
05:08is
05:08is equal to 1, 3,
05:142. So hopefully, guys, you
05:15can see that now. If
05:18you've got the equation of
05:18a plane, the normal is
05:20just 1, 3, 2. It's
05:22the A, B, C, and
05:24the A, X plus B,
05:25Y plus C, Z equation.
05:28So this is my normal.
05:29So this angle here, this
05:31is the right angle, this
05:32is phi, and this is
05:34theta.
05:36That's phi. So the cos
05:40cos of phi is equal
05:44to this dot this. So
05:46it's the direction vector for
05:481 negative 2 dot 1
05:523 2. So the direction
05:56vector or the normal direction
05:58vector dot the normal divided
06:01by x, this is just
06:03the
06:04the cosine angle formula divided
06:08by the magnitude, the magnitude
06:11of the direction vector, which
06:14is 4 squared plus 1
06:16squared plus 2 squared times
06:19the magnitude of 1 squared
06:24plus 3 squared plus 2
06:28squared. And this equals, let's
06:31do it here, this dot
06:32this is
06:33four, four, plus three, minus
06:38four is three. So it's
06:40three over the square root
06:45of, and I don't need
06:47that much, 16, 17, 21,
06:5121, the square root of
06:54nine, 10, 14. This is
06:56cos of theta, no sorry,
06:58this is cos of five,
06:59be careful. So then five,
07:01is equal to phi, which
07:05I'll have proper phi, phi
07:07is equal to the inverse
07:09cos, so I need to
07:10do this menu. That's our
07:12trig. Let me get out
07:13of that. Trig inverse cos,
07:17and I'm going to put
07:18this in just like I
07:19see it, so I have
07:20this, I have three over
07:24the square root of 21
07:26multiplied by
07:29by the square root of
07:3014. Press enter and I
07:34get 79 .9235 .79 .9235.
07:43That's 5. Therefore, theta equals
07:4890 minus 79 .9235, which
07:54equals, I'm just going
07:57going to go minus 90
07:58forgive me for this guys
07:59but I think that's I'm
08:00okay to do that it's
08:02obviously positive so 10 .0765
08:0510 .0765 degrees that is
08:11the angle that is the
08:13acute angle between the line
08:14and the plane the up
08:17to angle would obviously be
08:18180 minus 10 .0765 degrees
08:22okay hope that makes sense
08:25Definitely, definitely, definitely, this is
08:27one of those situations where
08:28you're much better off having
08:30a good understanding about what's
08:31going on than just learning
08:34the rule. If you can
08:36replicate this diagram in an
08:39exam, it will really set
08:40your mind at ease, because
08:42you can show the examiner
08:44that you know exactly what
08:45you're doing. Okay, I hope
08:47you understood everything that lesson.
08:49If not, let me know,
08:50and I'll see you in
08:51the next video.
08:53you