Video Transcript
Fill in the blank: If vectors 𝐀 one, three and 𝐁 three, 𝑥 are perpendicular, then 𝑥 is equal to what.
In this question, we’re given two vectors in terms of their components, the vector 𝐀 and the vector 𝐁. We’re told that these two vectors are perpendicular, and we need to use this to determine the value of 𝑥. To find the value of 𝑥, let’s start by recalling what it means for two vectors to be perpendicular. We say that two vectors are perpendicular if the dot product between these two vectors is equal to zero. In other words, if 𝐮 and 𝐯 are perpendicular, their dot product must be equal to zero. And if the dot product between 𝐮 and 𝐯 is equal to zero, then we can say that these vectors are perpendicular.
Therefore, because we’re told vectors 𝐀 and 𝐁 are perpendicular, the dot product between these two vectors must be equal to zero. We can use this to find the value of 𝑥. We’ll start by substituting the expressions we’re given for vector 𝐀 and vector 𝐁 in the question. We get the dot product between the vector one, three and the vector three, 𝑥 must be equal to zero. We then want to evaluate the left-hand side of this expression. And to do this, we need to remember to take the dot product of two vectors, we need to find the sum of the product of the corresponding components of the vectors.
If we take the product of the first component of each vector, we get one multiplied by three. If we take the products of the second component of each vector, we get three multiplied by 𝑥. The dot product is then the sum of these two values. And because these vectors are perpendicular, we know this must be equal to zero. We can then simplify the left-hand side of this equation. We get that three plus three 𝑥 must be equal to zero. We then want to solve this equation for 𝑥. We’ll start by subtracting three from both sides of the equation, giving us that three 𝑥 is equal to negative three. Then we just divide both sides of our equation through by three. We get three 𝑥 over three is equal to negative three divided by three which simplifies to give us that 𝑥 is equal to negative one, which is our final answer.
Therefore, we were able to show if the vectors 𝐀 one, three and 𝐁 three, 𝑥 are perpendicular, then the value of 𝑥 must be equal to negative one.
