In Exercises 23–32, use the dot product to determine whether v...

Question

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.

v = 3i, w = -4i

Accepted Answer

Hello. Today we're going to be determining whether the vectors A and B are orthogonal. So what does it mean for two vectors to be orthogonal to each other? Well, if we take the dot product of the two given vectors and the dot product is equal to zero, this will prove that the two vectors are orthogonal to each other. So what we'll need to do is we'll need to calculate the dot product of A and B vector A is given to us as 18 I. But notice that the vector A is missing the J component, we can represent the J component with a coefficient of zero and rewrite the A vector as 18 I plus zero J. And the same can be said for vector B vector B is missing the J component. So we can give the J component A coefficient of zero and rewrite the vector B as negative seven I plus zero J. Now, in order to calculate the dot product of A and B, we're going to be taking the product of the I components and J components together and taking the sum of those two products. So the dot product am dot B will be represented as 18 multiplied by negative seven plus zero, multiplied by zero 18, multiplied by negative seven is going to give us the value of negative 126 and zero multiplied by zero will just give us zero. Finally negative 126 plus zero is going to give us the value of negative 126. And since the dot product of A and B is not equal to zero, this proves 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.

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4i

Key Concepts

Dot Product

Orthogonality of Vectors

Vector Components and Notation

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