### Video Transcript

Given that ๐ด๐ต๐ถ is an isosceles triangle, where ๐ด๐ต is equal to ๐ด๐ถ is equal to six centimeters and the measure of angle ๐ด is 120 degrees, determine the scalar or dot product of vector ๐๐ with vector ๐๐.

Letโs begin by sketching the isosceles triangle. Weโre given that the measure of angle ๐ด is 120 degrees and that sides ๐ด๐ต and ๐ด๐ถ both have a length of six centimeters. Weโre asked to find the dot product of the vectors ๐๐ and ๐๐. And we know that the scalar or dot product of two vectors ๐ฎ and ๐ฏ is the product of their magnitudes with the cosine of the angle between them. So in our case this is the product of the magnitude of vector ๐๐ with the magnitude of vector ๐๐ and the cosine of the angle between them.

Now we know the magnitude of vector ๐๐, and thatโs six centimeters. So weโll need to find the magnitude of vector ๐๐ and the angle ๐ between the two vectors. To find the angle ๐, we recall that the angle between two vectors is defined uniquely as the angle between their directions when the lines representing them either both converge or both diverge. In the case of our vectors ๐๐ and ๐๐, we see that ๐๐ converges onto the vertex ๐ถ, whereas ๐๐ diverges from the vertex ๐ถ. And so to specify the angle ๐, we must extend ๐๐ so that both vectors diverge from the vertex ๐ถ. Our angle ๐ then is the obtuse angle between the two vectors.

Now, to find the measure of the angle ๐, we refer back to the fact that our triangle is an isosceles triangle. And this means that the measures of angles ๐ถ and ๐ต are the same. And since the angles in a triangle must sum to 180 degrees, we have the measures of angles ๐ด, ๐ต, and ๐ถ sum to 180. And since the measures of angles ๐ต and ๐ถ are the same, we have 180 is equal to the measure of angle ๐ด plus two times the measure of angle ๐ต. And this, of course, is the same as the measure of angle ๐ด plus two times the measure of angle ๐ถ.

And since the measure of angle ๐ด is 120 degrees, we can subtract 120 from both sides. And we have 60 degrees is two times the measure of angle ๐ต. And finally, dividing through by two, we have the measure of angle ๐ต is 30 degrees. And of course this is the same as the measure of angle ๐ถ.

So now making some space and making a note of the measure of the angles at ๐ต and ๐ถ, we can use this to find the measure of the angle ๐ by noting that ๐ plus the measure of angle ๐ถ must equal 180 degrees. And this is because the vectors ๐๐ and ๐ฎ lie on a straight line and the angle between them is 180 degrees. Since we just found that the measure of angle ๐ถ is 30 degrees, subtracting 30 from both sides, we have the measure of our angle ๐ is 150 degrees.

So now making a note of this, our next step is to find the magnitude of the vector ๐๐, that is, its length in centimeters, since the lengths of ๐ด๐ต and ๐ด๐ถ are both in centimeters. And to find the length ๐ต๐ถ, weโre gonna use the sine rule. This tells us that for triangle ๐ด๐ต๐ถ, the length of side ๐ over the sin of the angle ๐ด is the length of side ๐ over the sin of the angle ๐ต is the length of side ๐ over the sin of the angle at ๐ถ. And applying this to our triangle, we have six over the sin of 30 degrees is equal to ๐ต๐ถ over the sin of 120 degrees. And now multiplying through by the sin of 120 degrees, we have six times the sin of 120 degrees over the sin of 30 degrees is equal to ๐ต๐ถ.

Now since the sin of 120 degrees is root three over two and the sin of 30 degrees is one-half, we have six times the square root of three over two divided by one-half. Dividing by two on the top leaves us with three root three. And dividing by a half is the same as multiplying by two. And so we have our length ๐ต๐ถ is equal to six root three centimeters.

Now making some space so we can calculate our dot product, we have everything we need to find the dot product of the vectors ๐๐ and ๐๐, that is, six, which is the magnitude of vector ๐๐, multiplied by six root three, thatโs the magnitude of vector ๐๐, multiplied by the cos of 150 degrees, thatโs the cos of our angle ๐. The cos of 150 degrees is negative root three over two.

We can then cancel the two in our denominator with one of the sixes to give three, which leaves us with 18 multiplied by root three multiplied by negative root three. And finally this evaluates to negative 54. The dot or scalar product of vectors ๐๐ and ๐๐ is therefore equal to negative 54.