Determine whether each pair of vectors is orthogonal.
√5...

Question

Determine whether each pair of vectors is orthogonal.

√5i - 2j, -5i + 2 √5j

Accepted Answer

Hello. Today we are going to be showing whether the two given vectors are orthogonal or not. We are given the vector, the square root of 17 I minus eight J and we are given another vector negative 11 I plus four multiplied by the square root of 17 J. In order to show that the two vectors are orthogonal, we want to take the dot product of the two vectors and show that the dot product is equal to zero. If the dot product of the two vectors are equal to zero, then the vectors are orthogonal. If they are equal to any value that is not zero, then the vectors are not orthogonal. We are going to set U equal to the first vector, the square root of 17 I minus eight J. And we are going to take the second vector V and set it equal to negative 11 I plus four multiplied by the square root of 17 J. We can rewrite both of the given vectors in the unit component form. So we can write the vector U as the square root of 17 comma negative eight. And we can rewrite the vector V as negative 11 comma four multiplied by the square root of 17. Now, what we need to do is we need to take the dot product of both of these vectors. In order to take the dot product of both of the vectors, we are going to take the product of both of their X components, which is going to be the square root of 17 multiplied by negative 11 and add the product of the Y components, which is going to be negative eight multiplied by four multiplied by the square root of 17. Now let's go ahead and simplify the square root of 17 multiplied by negative 11 will give us negative 11 multiplied by the square root of 17, negative eight multiplied by four multiplied by the square root of 17 will leave us with negative 24 multiplied by the square root of 17. One multiplied by negative 24 multiplied by the square root of 17 will leave us with the negative value. So we have negative 11 multiplied by the square root of 17 minus 24 multiplied by the square root of 17. Finally, if we subtract the two terms together, we will be left with negative 35 multiplied by the square root of 17. And this value is not equal to zero, which shows that the two vectors are not orthogonal to each other. And with that being said, 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.
√5i - 2j, -5i + 2 √5j

Key Concepts

Vector Components and Notation

Dot Product of Vectors

Orthogonality of Vectors

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