Determine whether each pair of vectors is orthogonal.

Question

i + 3√2j, 6i - √2j

Accepted Answer

Hello, today we are going to be showing whether the two given vectors are orthogonal to each other. We are given the vector three I plus six multiplied by the square root of seven J. We are also given a second vector 13 I minus the square root of seven J. So how can we show that the two vectors are orthogonal to each other? Well, if we take the dot product of both of the vectors and show that the dot product is equal to zero. This will prove that both of the vectors are orthogonal to each other. If the dot product is equal to any other value that is not zero, then the vectors are not orthogonal to each other. Let's go ahead and let vector U equal to the first given vector which is three I plus six multiplied by the square root of seven J. We are also going to let the vector V equal to the second given vector which is 13 I minus the square root of seven J. We can rewrite both of the vectors in unit component form by using the coefficients of the I and J terms. So we can represent the vector U to be three comma six multiplied by the square root of seven. And we could represent the vector V as 13 comma negative square root of seven. Now, what we want to do is we want to take the dot product of both of these vectors. In order to take the dot product of U and V, we are going to multiply the X components of both of the vectors which will be three and 13 and add the product of the Y components of the vectors which will be six multiplied by the square root of seven multiplied by negative square root of seven three, multiplied by 13 will give us the value of 39 and six multiplied by the square root of seven multiplied by negative square root of seven will leave us with negative six multiplied by seven, negative six, multiplied by seven will give us the value of negative 42. So we have 39 plus negative 42 39 plus negative 42 will give us a final sum of negative three, but negative three is not equal to zero. So that means both of the given vectors are not orthogonal to each other. And the answer to this problem is going to be B So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Determine whether each pair of vectors is orthogonal.
i + 3√2j, 6i - √2j

Key Concepts

Vector Components and Representation

Dot Product of Vectors

Orthogonality of Vectors

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