WEBVTT
00:00:00.273 --> 00:00:15.363
Determine, to the nearest hundredth, the component of vector π along ππ given that π equals negative seven, two, 10 and the coordinates of π and π are one, negative four, negative eight and three, two, zero, respectively.
00:00:16.073 --> 00:00:22.633
Okay, in this exercise, we have a three-dimensional vector π and points in three-dimensional space π΄ and π΅.
00:00:23.033 --> 00:00:26.593
Letβs say that point π΄ is located here and point π΅ is here.
00:00:26.943 --> 00:00:31.913
We want to solve for the component of this given vector π along a vector ππ.
00:00:32.283 --> 00:00:37.233
This vector ππ will go from point π΄ to point π΅ looking like this.
00:00:37.593 --> 00:00:43.733
And to calculate the component of vector π along ππ, weβll want to know the components of vector ππ.
00:00:44.243 --> 00:00:50.333
To solve for those, we can subtract the coordinates of point π΄ from the coordinates of point π΅.
00:00:50.803 --> 00:00:55.543
In other words, we could write that vector ππ equals π minus π in vector form.
00:00:55.993 --> 00:01:11.843
Substituting in the coordinates of π΅ and π΄, we find subtracting those of π΄ from those of π΅ gives us a vector with components of three minus one or two, two minus negative four or six, and zero minus negative eight or eight.
00:01:13.013 --> 00:01:15.283
So then we now have our vector ππ.
00:01:15.513 --> 00:01:20.063
And as weβve seen, we want to solve for the component of vector π that lies along ππ.
00:01:20.583 --> 00:01:31.733
We can begin to do this by recalling that the scalar projection of one vector onto another is equal to the dot product of those two vectors divided by the magnitude of the vector being projected onto.
00:01:32.333 --> 00:01:41.113
In our example, as we calculate the component of vector π along ππ, weβre computing the scalar projection of π onto ππ.
00:01:41.713 --> 00:01:49.083
Therefore, we can say that the quantity we want to solve for is given by π dot ππ over the magnitude of vector ππ.
00:01:49.603 --> 00:02:01.283
Remembering that the magnitude of a vector is equal to the square root of the sum of the squares of the components of that vector, we see that what we want to calculate is this dot product over this square root.
00:02:01.643 --> 00:02:07.573
Carrying out this dot product, we start by multiplying the respective components of these two vectors together.
00:02:08.063 --> 00:02:14.563
And then working in our denominator, we know that two squared is four, six squared is 36, and eight squared is 64.
00:02:14.803 --> 00:02:22.413
So our fraction simplifies to negative 14 plus 12 plus 80 divided by the square root of four plus 36 plus 64.
00:02:22.883 --> 00:02:26.343
This equals 78 over the square root of 104.
00:02:26.723 --> 00:02:32.093
And we could leave this as our answer, except that weβre told to determine this overlap to the nearest hundredth.
00:02:32.593 --> 00:02:37.923
If we enter this fraction on our calculator then, to the nearest hundredth, it equals 7.65.
00:02:38.343 --> 00:02:41.703
Thatβs the component of vector π along vector ππ.