Question Video: Finding the Dot Product of Vectors Mathematics

Video Transcript

Suppose vector 𝚨 is equal to negative one, two, sevens, the magnitude of vector 𝚩 is equal to 13, and the angle between the two vectors is 135 degrees. Find the dot product of vector 𝚨 and vector 𝚩 to the nearest hundredth.

We can calculate the dot product of vector 𝚨 and vector 𝚩 by finding their magnitudes, multiplying these together, and then multiplying this by the cos of angle 𝜃, where 𝜃 is the angle between the two vectors. In this question, we are told that the magnitude of vector 𝚩 is equal to 13. And the angle between the two vectors is 135 degrees. In order to calculate the dot product of vector 𝚨 and vector 𝚩, we firstly need to calculate the magnitude of vector 𝚨.

Vector 𝚨 is written in terms of its three components, which we will call 𝑎 sub one, 𝑎 sub two, and 𝑎 sub three. The magnitude of any vector is a scalar quantity. It is equal to the square root of 𝑎 sub one squared plus 𝑎 sub two squared plus 𝑎 sub three squared. In this question, we begin by squaring negative one, two, and seven. These are equal to one four and 49, respectively. One plus four plus 49 is equal to 54. Therefore, the magnitude of vector 𝚨 is equal to the square root of 54.

Using the laws of radicals or surds, this can be rewritten as the square root of nine multiplied by the square root of six, as nine multiplied by six is 54. The square root of nine is equal to three. Therefore, the square root of 54 can be rewritten as three root six. This is the magnitude of vector 𝚨.

We can now calculate the dot product of vector 𝚨 and vector 𝚩. It is equal to three root six multiplied by 13 multiplied by the cos of 135 degrees. The cos of 135 degrees is equal to negative root two over two. This means that the dot product is equal to negative 39 root 12 over two. Root 12 can be rewritten as root four multiplied by root three. As root four is equal to two, this simplifies to negative 39 root three.

This isn’t the end of the question, however, as we are asked to give our answer to the nearest hundredth. Negative 39 root three is equal to negative 67.5499 and so on. Rounding this to two decimal places or the nearest hundredth is negative 67.55. The dot product of vector 𝚨 and vector 𝚩 to the nearest hundredth is negative 67.55.