In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. ...

Question

In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.

projᵤ (v + w)

Accepted Answer

Hello, today we're going to be using the given vectors to find the value of the given projection. So we are told that vector A is defined as three I minus eight J. We are also told that vector B is defined as two I plus six J and vector C is defined as negative eight J. So we want to use these three vectors to determine the projection of B plus C onto A. Now, normally, when we want to find a projection, the projection of A vector X onto Y is defined as the dot product of X and Y over the magnitude of Y squared multiplied by the vector of Y. But in this case here we have a sum of two vectors in the spot of X. So if we want to find the projection, we're going to be replacing every X value in the projection with B plus C. And we can rewrite the projection as the product or the dot product of B plus C dotted with A, all over the magnitude of A squared multiplied by the vector of A. So there are a few things we need in order to figure out our given projection first, let's determine the product or I'm sorry, the sum of B plus C. So we're going to be performing vector vector edition in order to figure out B plus C. So vector B is defined as two I plus six J and vector C is defined as negative eight J. We're going to add the two J components together and six J minus eight J will leave us with negative two J. Therefore, B plus C is defined as two I minus two J. Now that we have the sum of vector B and C, we'll, we need to find the dot product of B plus C dotted with A. So we'll need to find B plus C dotted with A and we know that B plus C is defined as two I minus two J. And again, vector A is defined as three I minus AJ. So in order to take the dot product, we'll need to multiply the I components together the J components together, then take the sum of the two products. So the dot product B plus C dotted with A will be defined as two multiplied by three plus negative two, multiplied by negative eight, three, multiplied by two will give us the value of six and negative two multiplied by negative eight will give us the value of 16. Finally, six plus 16 will give us the final value of 22. So we know that the dot Product B plus C dotted with A will equal to 22. The next thing we'll need for our projection is the magnitude of A squared. The magnitude of A squared is defined as the A I component squared, which is three squared plus the J component squared, which is negative eight squared three squared will give us the value of nine and negative eight squared will give us the value of 64. Finally, nine plus 64 will give us a final sum of 73. So now that we have the dot product B plus C dotted with A and we have the magnitude of A squared and we know our vector A, we can plug in all this information to figure out the projection of B plus C onto A. So the projection will equal to B plus C dotted with A which is 22 over the magnitude of A squared, which is 73 multiplied by the vector A which is three I minus eight J. In order to simplify this, we'll need to perform scalar multiplication onto the vector. And in order to do that, we'll distribute 22/73 into each element in vector A 22/73 multiplied by three will give us the value of 66/73 I and 22/73 multiplied it by negative eight will give us a value of negative 176 over J this is going to be the projection of B plus C onto A. And with that being said, the answer to this problem is going to be D. 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 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
projᵤ (v + w)

Key Concepts

Vector Addition

Vector Projection

Dot Product of Vectors

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