### Video Transcript

Given that ๐ด๐ต๐ถ is an isosceles triangle, where the side length ๐ด๐ต is equal to the side length ๐ด๐ถ is equal to 19 centimeters and the measure of angle ๐ด is equal to 51 degrees, determine the dot product between the vector ๐๐ and the vector ๐๐ correct to the nearest hundredth.

In this question, weโre given some information about triangle ๐ด๐ต๐ถ. First, weโre told that this represents an isosceles triangle. Next, weโre told the two equal sides of our isosceles triangle are the sides ๐ด๐ต and ๐ด๐ถ, which are 19 centimeters long. And weโre also told the measure of the angle at ๐ด is 51 degrees. We need to use this to determine the dot product between vector ๐๐ and vector ๐๐. And we need to give our answer to the nearest hundredth.

To answer this question, itโs always a good idea to sketch the information weโre given. So to start, we sketch an isosceles triangle ๐ด๐ต๐ถ with side ๐ด๐ต and side ๐ด๐ถ equal to 19 centimeters and the angle at ๐ด 51 degrees. Letโs now also sketch our two vectors ๐๐ and ๐๐ onto this diagram. ๐๐ will be the vector which starts at ๐ต and ends at ๐ด, and ๐๐ will be the vector which starts at ๐ต and ends at ๐ถ. So now we can see on our sketch the vector ๐๐ and the vector ๐๐ which weโre asked to calculate the dot product of.

And we can notice something interesting about these two vectors. We can actually calculate the angle between them. And we actually know a result connecting the dot product of two vectors with the angle between them. So we can use this to evaluate our dot product. We recall if ๐ is the angle between two vectors ๐ฎ and ๐ฏ, then the cos of ๐ must be equal to the dot product between ๐ฎ and ๐ฏ divided by the modulus of ๐ฎ times the modulus of ๐ฏ. And we know some of this information for the dot product given to us in the question.

For example, we can rewrite this formula with the vectors ๐๐ and the vectors ๐๐. We can also include the angle between these vectors on our diagram. Weโll call this ๐. We then have the cos of ๐ is equal to the dot product between vector ๐๐ and vector ๐๐ divided by the modulus of ๐๐ times the modulus of ๐๐. So to find the dot product using this method, weโre going to need to find the value of ๐ and weโre also going to need to find the modulus of vector ๐๐ and vector ๐๐.

We can see one of these straightaway. The modulus of vector ๐๐ is going to be the length of the side. Itโs going to be 19. Next, letโs find the value of ๐. To do this, weโre going to need to use the fact that triangle ๐ด๐ต๐ถ is an isosceles triangle. And we know in an isosceles triangle, the angles opposite the equal sides are equal. So our other unknown angle is also equal to ๐. But this is not the only thing we know. We also know the sum of the interior angles in a triangle adds to 180 degrees. So we can add these three angles together to give us that 180 degrees should be equal to ๐ plus ๐ plus 51 degrees.

Now we solve for ๐. First, weโll subtract 51 degrees from both sides of the equation and then simplify. This gives us 129 degrees is equal to two ๐. Then we just divide through by two, giving us that 64.5 degrees is equal ๐. So the last unknown in our equation is the magnitude of vector ๐๐. And this is going to be equal to the length of the side of our triangle ๐ต๐ถ. Usually, we represent this with a lowercase ๐ because itโs opposite the angle capital ๐ด. Thereโs several different ways we could find this length. For example, we know the opposite angle and the other two sides of our triangle, so we could do this by using the law of cosines.

Applying the law of cosines to this triangle to find our length gives us the modulus of ๐๐ all squared is equal to 19 squared plus 19 squared minus two times 19 multiplied by 19 times the cos of 51 degrees. And itโs also worth pointing out here we didnโt need to use the law of cosines. We couldโve also used the law of sines. Because weโve already found the value of ๐, we already know all the angles in our triangle. Itโs personal preference which one do we use. Weโll just use the law of cosines.

Evaluating the right-hand side of this equation, we get 267.63, and this continues. Then we can find the side length ๐ต๐ถ by taking the square root of both sides. We get the length of the side ๐ต๐ถ, or the modulus of vector ๐๐, is equal to 16.359, and this continues. Now all we need to do is substitute these values into our equation to find the dot product. We get the cos of 64.5 degrees is equal to the dot product between vector ๐๐ and vector ๐๐ divided by 19 times 16.359, and this expansion continues. And itโs important we donโt round this value until the very last step in our calculation.

Now to find our dot product, weโre going to need to multiply through by the denominator in our fraction. This gives us the dot product between vector ๐๐ and vector ๐๐ is equal to 19 times 16.359, and this continues, multiplied by the cos of 64.5. And if we calculate this expression, we get 133.815, and this expansion continues.

But remember the question wanted us to give our answer to the nearest hundredth. The nearest hundredth means two decimal places. So we need to look at our third decimal place to determine whether we round up or round down. We see the third decimal place is five, which is greater than or equal to five. This means we need to round up. And this gives us our final answer of 133.82. Therefore, we were able to show if ๐ด๐ต๐ถ is an isosceles triangle, where the side ๐ด๐ต is equal to the side ๐ด๐ถ is equal to 19 centimeters and the measure of angle ๐ด is 51 degrees, then the dot product between the vectors ๐๐ and ๐๐ to the nearest hundredth will be 133.82.