Video Transcript
Determine, to the nearest
hundredth, the component of vector 𝐕 along 𝐀𝐁 given that 𝐕 equals negative
seven, two, 10 and the coordinates of 𝐀 and 𝐁 are one, negative four, negative
eight and three, two, zero, respectively.
Okay, in this exercise, we have a
three-dimensional vector 𝐕 and points in three-dimensional space 𝐴 and 𝐵. Let’s say that point 𝐴 is located
here and point 𝐵 is here. We want to solve for the component
of this given vector 𝐕 along a vector 𝐀𝐁. This vector 𝐀𝐁 will go from point
𝐴 to point 𝐵 looking like this. And to calculate the component of
vector 𝐕 along 𝐀𝐁, we’ll want to know the components of vector 𝐀𝐁.
To solve for those, we can subtract
the coordinates of point 𝐴 from the coordinates of point 𝐵. In other words, we could write that
vector 𝐀𝐁 equals 𝐁 minus 𝐀 in vector form. Substituting in the coordinates of
𝐵 and 𝐴, we find subtracting those of 𝐴 from those of 𝐵 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
𝐀𝐁. And as we’ve seen, we want to solve
for the component of vector 𝐕 that lies along 𝐀𝐁.
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 𝐕 along 𝐀𝐁, we’re computing the scalar projection of 𝐕 onto
𝐀𝐁. Therefore, we can say that the
quantity we want to solve for is given by 𝐕 dot 𝐀𝐁 over the magnitude of vector
𝐀𝐁.
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 we’re 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. That’s the component of vector 𝐕
along vector 𝐀𝐁.
