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Question Video: Finding the Component of a Vector Mathematics • 12th Grade

Find the component of vector 𝚨 in the direction of 𝚩, where πœƒ is the included angle between them.

01:49

Video Transcript

Find the component of vector 𝚨 in the direction of vector 𝚩, where πœƒ is the included angle between them.

We begin by sketching the three-dimensional coordinate plane. We can add on the arbitrary vector 𝚨 as shown, where vector 𝚨 has components 𝐴 sub π‘₯, 𝐴 sub 𝑦, and 𝐴 sub 𝑧. We are asked to find the component of this vector in the direction of vector 𝚩, where vector 𝚩 has components 𝐡 sub π‘₯, 𝐡 sub 𝑦, and 𝐡 sub 𝑧. This is in effect the scalar projection of vector 𝚨 onto vector 𝚩. And we know that this is equal to the dot or scalar product of vectors 𝚨 and 𝚩 divided by the magnitude of vector 𝚩.

From the definition of the dot or scalar product, we can rewrite the numerator of our expression as the magnitude of 𝚨 multiplied by the magnitude of 𝚩 multiplied by cos πœƒ. Since the magnitude of vector 𝚩 cannot equal zero, we can cancel this from the numerator and denominator. And this leaves us with the magnitude of vector 𝚨 multiplied by cos πœƒ. This is the component of vector 𝚨 in the direction of vector 𝚩, where πœƒ is the included angle between them. And we can see this directly from our diagram by using our knowledge of right-angled trigonometry.

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