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Math  /  Algebra

QuestionFind <3uˉ,4vˉ+5wˉ><3 \bar{u}, 4 \bar{v}+5 \bar{w}>, given that ⟨uˉ,vˉ>=6,<vˉ,wˉ>=−5\langle\bar{u}, \bar{v}>=6,<\bar{v}, \bar{w}>=-5, and <uˉ,wˉ>=−1<\bar{u}, \bar{w}>=-1 Number

Studdy Solution

STEP 1

1. We are given vectors uˉ\bar{u}, vˉ\bar{v}, and wˉ\bar{w}.
2. The notation <⋅,⋅><\cdot,\cdot> represents the dot product of two vectors.
3. We need to find the dot product <3uˉ,4vˉ+5wˉ><3 \bar{u}, 4 \bar{v} + 5 \bar{w}>.

STEP 2

1. Distribute the scalar multiplication inside the dot product.
2. Apply the distributive property of the dot product.
3. Use the given dot product values to compute the result.

STEP 3

Start by distributing the scalar multiplication inside the dot product:
<3uˉ,4vˉ+5wˉ>=3<uˉ,4vˉ+5wˉ> <3 \bar{u}, 4 \bar{v} + 5 \bar{w}> = 3 <\bar{u}, 4 \bar{v} + 5 \bar{w}>

STEP 4

Apply the distributive property of the dot product:
3<uˉ,4vˉ+5wˉ>=3(4<uˉ,vˉ>+5<uˉ,wˉ>) 3 <\bar{u}, 4 \bar{v} + 5 \bar{w}> = 3 (4 <\bar{u}, \bar{v}> + 5 <\bar{u}, \bar{w}>)

STEP 5

Substitute the given values of the dot products into the equation:
3(4⋅6+5⋅(−1)) 3 (4 \cdot 6 + 5 \cdot (-1))

STEP 6

Calculate the expression inside the parentheses:
3(24−5)=3×19 3 (24 - 5) = 3 \times 19

STEP 7

Compute the final result:
3×19=57 3 \times 19 = 57
The solution is:
57 \boxed{57}

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