Video Transcript
Find the component of vector π¨ in the direction of vector π©, where π is the
included angle between them.
We begin by sketching the three-dimensional coordinate plane. We can add on the arbitrary vector π¨ as shown, where vector π¨ has components π΄ sub
π₯, π΄ sub π¦, and π΄ sub π§. We are asked to find the component of this vector in the direction of vector π©,
where vector π© has components π΅ sub π₯, π΅ sub π¦, and π΅ sub π§. This is in effect the scalar projection of vector π¨ onto vector π©. And we know that this is equal to the dot or scalar product of vectors π¨ and π©
divided by the magnitude of vector π©.
From the definition of the dot or scalar product, we can rewrite the numerator of our
expression as the magnitude of π¨ multiplied by the magnitude of π© multiplied by
cos π. Since the magnitude of vector π© cannot equal zero, we can cancel this from the
numerator and denominator. And this leaves us with the magnitude of vector π¨ multiplied by cos π. This is the component of vector π¨ in the direction of vector π©, where π is the
included angle between them. And we can see this directly from our diagram by using our knowledge of right-angled
trigonometry.
