Let u = 〈-2, 1〉, v = 〈3, 4〉,...

Question

Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.

u • v - u • w

Accepted Answer

Hello. Today we are going to be using the given vectors to find the value of U dot V plus U dot W. We are told that the vector U is equal to negative 10 comma six. Vector V is equal to 12 comma 19 and vector W is equal to 15 comma four. So how can we use these vectors to find the value of U dot V plus U dot W? Well, we need to recall the definition of the dot product of two vectors. Let's say we have two generic vectors. We have a U vector which is equal to X one comma Y one and we have a generic vector V which is equal to X two, Y two. The dot product U dot V is equal to the product of the X values. So X one multiplied by X two plus the product of the Y values Y one dot Y two. Using this definition, we will first need to find the values of UV and UW. Let's go ahead and begin this process by solving for you dot V. So we are told that U is equal to negative 10 comma six and V is equal to 12 comma 19 UV will equal to the product of the X values which is negative 10 multiplied by 12 plus the product of the Y values which will be six and 19, negative 10, multiplied by 12 will give us negative 126 multiplied by 19 will leave us with the value of 114 one H negative 120 plus 114 will give us a final value of negative six. So we know that the value of U dot V is equal to negative six. Now we need to solve for you dot W. Now we know that the X component of the U vector is negative 10 and the X component of the W vector is 15. So the dot product will equal to negative 10 multiplied by 15 plus the product of the Y components which are going to be six and four, negative 10, multiplied by 15 will leave us with the value of negative 156 multiplied by four will give us 24 negative 150 plus 24 will give us the value of negative 126. So now we have the value of U dot W now that we have both of the dot products we can solve for the value of U dot V plus U dot W. We can substitute you dot V to be negative six and we can substitute you dot W to be negative 126 negative six plus negative 126 will give us a final value of negative 132. And this is going to be the final value of U dot V plus U dot W 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.

Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.
u • v - u • w

Key Concepts

Dot Product of Vectors

Vector Components and Notation

Properties of the Dot Product

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