00:00Hi guys, so in this
00:02video we're just gonna look
00:03at the properties of the
00:05vector product. So we had
00:08this in the previous lesson.
00:09V cross W is equal
00:10to negative W cross V.
00:12So it goes the, it's
00:14the same magnitude but it
00:16goes the opposite direction. U
00:21cross V plus W is
00:23just equal to U cross
00:24V plus U cross W.
00:25That's nice. And then when
00:26you multiply by K as
00:28scalar kv cross w is
00:29equal to k times v
00:31cross w. That's nice. And
00:32then v cross v is
00:33equal to zero. That's, well,
00:36you can think of that
00:37because the sign of the
00:38angle between two vectors, well,
00:42the angle between two vectors
00:43that are the same is
00:44zero. So you end up
00:46multiplying it by sign of
00:47zero, which gives you zero.
00:51Okay, let's just do a
00:52few of these questions.
00:56There weren't many when I
00:57looked at the past papers.
00:58There's not many of these
00:59types of questions, but certainly
01:00you do need to be
01:03comfortable with dealing with anything
01:04like this. So question one,
01:06show that a minus b
01:08cross a plus b is
01:09equal to two a cross
01:12b. So I can just
01:13multiply these out kind of
01:14like this. So I can
01:16say this is a. So
01:18let's write the whole thing
01:19here. A minus b cross
01:23a
01:24plus b. So this is
01:26equal to a cross a,
01:27a cross a, plus a
01:33cross b, plus a cross
01:37b, minus b cross a,
01:41minus b cross a, minus
01:45b cross b.
01:52Now what's a cross a?
01:55Well, look at it right
01:55here. It's zero. And what's
01:57b cross b? Well, that's
01:59zero. So what I'm left
02:00with is, so this equals,
02:03this is zero. This is
02:05zero. And I have a
02:06cross b, a cross b
02:10minus b cross a. Now
02:13this is actually equal to,
02:16from this, minus b cross
02:19a is
02:20is equal to a cross
02:21b. So this is actually
02:22equal to, let's just write
02:23the mod on zero plus
02:26a cross b plus a
02:28cross b minus zero, which
02:32is equal to two a
02:34cross b, two a cross
02:39b. And because of this,
02:41it doesn't matter, am I
02:41saying two a cross b,
02:43or am I saying two
02:45a cross b, it's the
02:47same thing.
02:48Okay, next one, question two.
02:53Simplify. Let's write it down.
02:56Two, A minus three, B
02:58cross, and three, A plus
03:03two B. This equals six,
03:07A cross A, that's gonna
03:08be zero, nice, and plus
03:11four, A cross B minus
03:15nine,
03:16b cross a, this is
03:21actually quite similar, minus 6
03:24b cross b. So this
03:26is equal to 0, that's
03:280, that's 0, and this
03:30is 4, this is very,
03:33very similar, 4 a cross
03:34b plus 9 a cross
03:37b, which is 13 a,
03:4013 a cross b. Last
03:44one.
03:44Evaluate A cross B dot
03:49A minus B. Okay, so
03:53A cross B dot A
04:00minus B is equal to,
04:02so remember this is just
04:03a vector. So this is
04:04a vector dot A minus
04:06B. So using the scalar
04:07product properties, I can say
04:09this is equal to A
04:11cross B
04:12dot a, and there's a
04:16minus here, so minus a
04:19cross b dot b. Now,
04:24if you remember, when you
04:28do a cross b, and
04:29this is like the most
04:31important property of the vector
04:33product, when you cross two
04:34vectors, when you do a
04:35cross b, you get a
04:37vector that is perpendicular to
04:39a and perpendicular
04:40to be. So here I've
04:43got a vector perpendicular to
04:45A. So what's the dot
04:47product of two perpendicular vectors?
04:49Well, it's zero. So that's
04:50zero minus. Again, here I
04:54have a vector perpendicular to
04:56B. So what's the dot
04:58product of two perpendicular vectors?
04:59That is also zero. So
05:00you just have it zero
05:01minus zero equals zero. Okay,
05:04that's it. Pretty short video,
05:06short lesson. And yes,
05:08I certainly want you to
05:11know these. You could be
05:13asked a question like this,
05:14no doubt about it. Hopefully
05:17it's fairly straightforward. The kind
05:19of main things remember are
05:20this v cross w equals
05:22negative w cross v and
05:24then v cross v equals
05:25zero. Okay, see you in
05:27the next lesson.