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

Fi...

Question

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

Find each specified scalar or vector.
5u ⋅ (3v - 4w)

Accepted Answer

Hello, today we're going to be determining the specified scaler using the given information. So what we are given is three vectors. We are told that vector A is equal to three I minus eight J. We are given a second vector B which is equal to two I plus six J and we are given a third vector C which is equal to negative eight J. We are then told that the specified scalar is given to us as two A multiplied by five B minus three C. So in order to find the scalar, let's first evaluate this little by little. We're going to start by evaluating three B, I'm sorry, five B minus three C. So what we're going to do is we're going to plug in the vectors B N C into this expression. Doing so gives us five multiplied by two, I plus six J minus three, multiplied by negative eight J. In order to simplify this, we're going to distribute five into the vector two I plus six J and we're going to distribute negative three with the vector negative eight J. Doing so leaves us with 10 I plus 30 J plus 24 J. Then what we're going to do is we're going to add the J components together 30 plus 24 will give us the value of 54. And the inside portion five B minus three C is going to simplify to the vector 10 I plus 54 J. Next, let's go ahead and evaluate two A two A is going to be the scalar multiplication of two and the vector A and the vector A is given to us as three I minus A to J. So in order to evaluate this, we're going to distribute the scale or two into the vector three I minus eight J. And that is going to leave us with six I minus 16 J. So if we were to res substitute this into the original expression, the original expression can be rewritten as the vector six I minus 16 J multiplied by 10 I plus 54 J. Now, in order to take the product of two vectors, we're going to need to evaluate the dot product of the two vectors. The dot product is the sum of the products of the individual components. So what this means is that we're going to be multiplying the I components together and we're gonna multiply the J components together and take the sum of the products. So the dot product of these two vectors is going to be six multiplied by 10 plus negative 16, multiplied by 54 six multiplied by 10 will give us the value of 60 and negative 16 multiplied by 54 will give us the value of negative 864 60 minus negative. I'm sorry, 60 minus 8 64 will leave us with the value of negative 804. And this is going to be the value of the specified scaler. And with that being said, 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 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector. 5u ⋅ (3v - 4w)

Key Concepts

Vector Operations

Dot Product

Scalar Multiplication

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