Question Video: Determining the Orthogonality of Two Given Vectors

True or false: vectors ๐€ = <โˆ’3, 1> and ๐ = <โˆ’2, โˆ’6> are perpendicular. [A] True [B] False


Video Transcript

True or false: vectors ๐€ negative three, one and ๐ negative two, negative six are perpendicular. Option (A) true or option (B) false.

In this question, weโ€™re given two vectors in terms of components, the vector ๐€ and the vector ๐. We need to determine if these two vectors are perpendicular. To answer this question, letโ€™s first recall 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 ๐ฎ dot ๐ฏ is equal to zero, then vectors ๐ฎ and ๐ฏ are perpendicular. This means we can check if vectors ๐€ and ๐ are perpendicular by evaluating the dot product of these two vectors. If this evaluates to give us zero, the vectors are perpendicular. Otherwise, we can say they are not perpendicular. Weโ€™ll start by substituting the expressions weโ€™re given for vectors ๐€ and ๐ in the question.

We need to find the dot product between the vector negative three, one and the vector negative two, negative six. Remember, to evaluate the dot product of two vectors, we need to find the sum of the product of the corresponding components of the two vectors. We can start by taking the products of the first components of the two vectors. Thatโ€™s negative three multiplied by negative two. We can also multiply the second component of the two vectors together. Thatโ€™s one multiplied by negative six. And then the dot product is the sum of these two values. We can then evaluate each of these terms. Negative three multiplied by negative two is equal to six, and one times negative six is equal to negative six. This gives us six plus negative six, which is the same as six minus six, which is of course just equal to zero.

Therefore, weโ€™ve shown the dot product between vector ๐€ and vector ๐ is equal to zero, which is exactly the same as saying the two vectors are perpendicular. Therefore, to answer the question โ€œIs it true or false the vector ๐€ negative three, one and the vector ๐ negative two, negative six are perpendicular?โ€ we showed that this is true. They are, in fact, perpendicular.

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