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Math  /  Algebra

Question- Question 3
1 point Number Help
Find <5uˉ,5vˉ+2wˉ><5 \bar{u}, 5 \bar{v}+2 \bar{w}>, given that <uˉ,vˉ>=4,<vˉ,wˉ>=−5<\bar{u}, \bar{v}>=4,<\bar{v}, \bar{w}>=-5, and <uˉ,wˉ>=−2<\bar{u}, \bar{w}>=-2 Number

Studdy Solution

STEP 1

1. The notation <aˉ,bˉ> <\bar{a}, \bar{b}> represents the dot product of vectors aˉ \bar{a} and bˉ \bar{b} .
2. We are given the dot products <uˉ,vˉ>=4 <\bar{u}, \bar{v}> = 4 , <vˉ,wˉ>=−5 <\bar{v}, \bar{w}> = -5 , and <uˉ,wˉ>=−2 <\bar{u}, \bar{w}> = -2 .
3. We need to find the dot product <5uˉ,5vˉ+2wˉ> <5 \bar{u}, 5 \bar{v} + 2 \bar{w}> .

STEP 2

1. Distribute the dot product over vector addition.
2. Use the scalar multiplication property of dot products.
3. Substitute the given dot products and simplify.

STEP 3

Distribute the dot product over the addition of vectors:
<5uˉ,5vˉ+2wˉ>=<5uˉ,5vˉ>+<5uˉ,2wˉ> <5 \bar{u}, 5 \bar{v} + 2 \bar{w}> = <5 \bar{u}, 5 \bar{v}> + <5 \bar{u}, 2 \bar{w}>

STEP 4

Apply the scalar multiplication property of dot products:
<5uˉ,5vˉ>=5⋅5⋅<uˉ,vˉ> <5 \bar{u}, 5 \bar{v}> = 5 \cdot 5 \cdot <\bar{u}, \bar{v}> <5uˉ,2wˉ>=5⋅2⋅<uˉ,wˉ> <5 \bar{u}, 2 \bar{w}> = 5 \cdot 2 \cdot <\bar{u}, \bar{w}>

STEP 5

Substitute the given values of the dot products:
<5uˉ,5vˉ>=25⋅4=100 <5 \bar{u}, 5 \bar{v}> = 25 \cdot 4 = 100 <5uˉ,2wˉ>=10⋅(−2)=−20 <5 \bar{u}, 2 \bar{w}> = 10 \cdot (-2) = -20

STEP 6

Add the results from the previous step:
<5uˉ,5vˉ+2wˉ>=100+(−20)=80 <5 \bar{u}, 5 \bar{v} + 2 \bar{w}> = 100 + (-20) = 80
The value of the dot product is:
80 \boxed{80}

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