Question Video: Finding the Dot Product between Vectors | Nagwa Question Video: Finding the Dot Product between Vectors | Nagwa

# Question Video: Finding the Dot Product between Vectors Mathematics • Third Year of Secondary School

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The angle between π and π is 22Β°. If |π| = 3|π| = 25.2, find π β π to the nearest hundredth.

02:11

### Video Transcript

The angle between vector π and vector π is 22 degrees. If the magnitude of vector π is equal to three times the magnitude of vector π is equal to 25.2, find the dot product between π and π to the nearest hundredth.

In this question, weβre given some information about two vectors π and π. First, weβre told the angle between these two vectors is equal to 22 degrees. Next, weβre also told information about their magnitudes. We know the magnitude of π is equal to 25.2, and we know that three times the magnitude of π is also equal to 25.2. So the magnitude of π is three times bigger than the magnitude of π. We need to use this to find the dot product of π and π. And we need to give our answer to the nearest hundredth.

To answer this question, we need to notice that we know a formula which connects the angle between two vectors with their dot product. We recall if π is the angle between two vectors π and π, then we know that the cos of π must be equal to the dot product between π and π divided by the magnitude of π times the magnitude of π. And in this question, we already know some of these values. For example, weβre told the angle between our two vectors is 22 degrees. Next, weβre also told the magnitude of π is equal to 25.2.

And we could also find the magnitude of π using the information given to us in the question. One way of doing this is to notice that three times the magnitude of π is equal to 25.2. We can then solve this to find the magnitude of π by dividing both sides of our equation through by three. And calculating this, we get the magnitude of π is 8.4. So, in fact, the only unknown in this equation is the dot product between π and π. And thatβs exactly what weβre asked to calculate.

So weβll substitute the angle of π equal to 22 degrees, the magnitude of π equal to 25.2, and the magnitude of π equal to 8.4 into our equation. This gives us the cos of 22 degrees should be equal to the dot product between π and π divided by 25.2 times 8.4. And now we can just rearrange this equation for the dot product between π and π. We multiply through by 25.2 times 8.4. This gives us π dot π is 25.2 times 8.4 times the cos of 22 degrees. And we can calculate this to the nearest hundredth or to two decimal places. Itβs equal to 196.27.

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