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

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 the two given vectors are parallel or orthogonal to each other. So we are told that vector A is defined as 17 I minus 24 J. And we are told that vector B is defined as I minus 120 J. Let's go ahead and start by determining whether the two vectors are orthogonal or not. So in order to show whether two vectors are orthogonal, we need to show that their dot product is equal to zero. So we're going to take the dot product of A and B in order to take the dot product, we'll multiply the I components together multiply the J components together then take the sum of the two products. So the dot product will be defined as 17 multiplied by 85 plus negative 24 multiplied by, by negative 120 17, multiplied by 85 will give us the value of 1445 and negative multiplied by negative 120 will give us the value of positive 2880. Finally, the sum of these two values will give us the value of 4325. So what we just showed is that the dot product A dot B is not equal to zero. So this proves that the two vectors are not orthogonal to each other. Now, let's go ahead and check to see whether the two vectors are parallel to each other. In order to show whether two vectors are parallel or not, we need to show that the dot product is equal to the product of the two magnitudes of the vectors or in other words, the dot product A dot B has to equal to the magnitude of A multiplied by the magnitude of B. So let's go ahead and calculate the magnitude of A and magnitude of B. The magnitude of A vector is defined as the square root of the I component squared plus the J component squared. So if we want to find the magnitude of vector A, the magnitude of vector A will be defined as the square root of its I component, which is 17 squared plus the square of its J component, which is negative 24 squared, 17 squared will give us the value of and negative 24 squared will give us the value of 576. Finally, the sum of these two numbers will give us the value of 865. So the magnitude of A will equal to the square root of 865. Now, we're going to use the same method to calculate the magnitude of B, the magnitude of B will be defined as the square root of its I component squared, which is 85 squared plus its J component squared, which is negative 120 squared, 85 squared will give us the value of 7225 and negative squared will give us the value of 14,400. Finally, if we add these two values together, will get a final value of 21,625. So the magnitude of B will equal to the square root of 21,625. Now we're going to take the product of the two magnitudes. So the product of the magnitude of A multiplied by the magnitude of B will give us the square root of 865 multiplied by the square root of 21,625. And if we multiply these two square roots together and simplify, we end up with a final value of 4325. So we know that the product of the magnitudes is 4325. And we know that the dot product A dot B is equal to 4325. So this confirms that the magnitudes or the product of the magnitudes is equal to the dot product of A and B which proves that the two vectors are parallel to each other. And with that being said, the answer to this problem is going to be A. 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 Representation in Component Form

Parallel Vectors

Orthogonal Vectors and the Dot Product

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