### Video Transcript

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.

Okay, in this exercise, we have a
three-dimensional vector π and points in three-dimensional space π΄ and π΅. Letβs say that point π΄ is located
here and point π΅ is here. We want to solve for the component
of this given vector π along a vector ππ. This vector ππ will go from point
π΄ to point π΅ looking like this. And to calculate the component of
vector π along ππ, weβll want to know the components of vector ππ.

To solve for those, we can subtract
the coordinates of point π΄ from the coordinates of point π΅. In other words, we could write that
vector ππ equals π minus π in vector form. 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. So then we now have our vector
ππ. And as weβve seen, we want to solve
for the component of vector π that lies along ππ.

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. In our example, as we calculate the
component of vector π along ππ, weβre computing the scalar projection of π onto
ππ. Therefore, we can say that the
quantity we want to solve for is given by π dot ππ over the magnitude of vector
ππ.

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. Carrying out this dot product, we
start by multiplying the respective components of these two vectors together. And then working in our
denominator, we know that two squared is four, six squared is 36, and eight squared
is 64. So our fraction simplifies to
negative 14 plus 12 plus 80 divided by the square root of four plus 36 plus 64. This equals 78 over the square root
of 104.

And we could leave this as our
answer, except that weβre told to determine this overlap to the nearest
hundredth. If we enter this fraction on our
calculator then, to the nearest hundredth, it equals 7.65. Thatβs the component of vector π
along vector ππ.