In Exercises 45–50, determine whether v and w are parallel, ortho...

Question

In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither.

v = 3i - 5j, w = 6i + 10j

Accepted Answer

Hello. Today we're going to be determining whether vectors A and B are parallel orthogonal to each other or do not possess any of the properties. Now, vector A is given to us as 17, I minus 24 J and vector B is given to us as 85 I plus 120 J. Let's start by checking to see whether A and B are orthogonal to each other. Now, if two vectors are orthogonal to each other, that means the dot product of both of those vectors will be equal to zero. So let's start by calculating the dot product product of A and B. Now, in order to calculate the dot product of A and B, we're going to multiply the I components together multiply the J components together then take the sum of both of those products. So the dot product of A and B is equal to 17, multiplied by 85 plus negative 24 multiplied by 17, multiplied by 85 will give us the value of 1445 and negative multiplied by 100 and 20 will give us the final value of negative 2880. And the difference of these two numbers will give us negative 1435. So we've calculated that the dot product of A and B is equal to negative 1435 and this dot product is not equal to zero. What this means is that the vectors A and B are not orthogonal to each other. Next, we'll need to check to see whether the two vectors are parallel to each other. Now, in order to determine whether two vectors are parallel to each other, the dot product of those two vectors must equal the product of the magnitude of those vectors. So we already have the value of the dot product of A and B. What we'll need to do is we'll need to calculate the magnitude of A, the magnitude of B and take the product of both of those magnitudes. Let's start by taking the magnitude of A. Now the magnitude of A will be the square root of the sum of its components squared. And recall that the vector A is given to us as 17 I minus 24 J. That means that the magnitude will be defined as the square root of 17 squared plus negative squared, 17 squared will give us the value of and negative 24 squared will give us the value of 576 289 plus 576 will give us a final sum of 865. So the magnitude of A is equal to the square root of 865. Next, we'll need to calculate the magnitude of B and just like A, we're going to take the square root of the sum of its components squared. Now, vector B is given to us as 85 I plus 120 J. That means the magnitude of B is equal to the square root of squared plus 120 squared. 85 squared will give us a value of and 120 squared will give us the value of 14,400. The sum of both of these numbers will give us 21,625. So the magnitude of B is equal to the square root of 21,625. Now, what we'll need to do is we'll need to take the product of the magnitude of A and the magnitude of B that is going to be the product of the square root of 865 multiplied by the square root of 21,625. And if we take the product of both of these values, the final product will give us the value of 4325. Now recall that the dot product of A and B is equal to negative 1435. That means the product of the magnitudes A and B is not equal to the dot product of A and B. And that proves that both A and B are not parallel to each other. Now, since A and B are not parallel or orthogonal, that means the answer to this problem is going to be C. 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 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i + 10j

Key Concepts

Vector Operations

Dot Product

Parallel Vectors

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