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2-5. Working with vectors Let u=⟨3,−4⟩\mathbf{u}=\langle 3,-4\rangle and $\mathbf{v}=\...
Jan 25, 2024
2-5. Working with vectors Let u=⟨3,−4⟩\mathbf{u}=\langle 3,-4\rangle and v=⟨−1,2⟩\mathbf{v}=\langle-1,2\rangle. Evaluate each of the following. 2. u−v\mathbf{u}-\mathbf{v} 3. −3v-3 \mathbf{v} 4. u+2v\mathbf{u}+2 \mathbf{v} 5. 2v−u2 \mathbf{v}-\mathbf{u}
Solution by Steps
step 1
To find the vector u−v\mathbf{u}-\mathbf{v}, we subtract the corresponding components of v\mathbf{v} from u\mathbf{u}
step 2
Subtracting the vectors: (3,−4)−(−1,2)=(3−(−1),−4−2)(3, -4) - (-1, 2) = (3 - (-1), -4 - 2)
step 3
Simplifying the components: (3+1,−4−2)=(4,−6)(3 + 1, -4 - 2) = (4, -6)
step 4
The resulting vector is u−v=(4,−6)\mathbf{u}-\mathbf{v} = (4, -6)
Answer
u−v=(4,−6)\mathbf{u}-\mathbf{v} = (4, -6)
Key Concept
Vector Subtraction
Explanation
To subtract vectors, subtract corresponding components of the second vector from the first.
Solution by Steps
step 1
To find −3v-3\mathbf{v}, we multiply each component of v\mathbf{v} by −3-3
step 2
Multiplying the vector: −3(−1,2)=(−3⋅−1,−3⋅2)-3(-1, 2) = (-3 \cdot -1, -3 \cdot 2)
step 3
Simplifying the components: (3,−6)(3, -6)
step 4
The resulting vector is −3v=(3,−6)-3\mathbf{v} = (3, -6)
Answer
−3v=(3,−6)-3\mathbf{v} = (3, -6)
Key Concept
Scalar Multiplication of a Vector
Explanation
To multiply a vector by a scalar, multiply each component of the vector by the scalar.
Solution by Steps
step 1
To find u+2v\mathbf{u}+2\mathbf{v}, we first multiply v\mathbf{v} by 2 and then add it to u\mathbf{u}
step 2
Multiplying v\mathbf{v} by 2: 2(−1,2)=(2⋅−1,2⋅2)2(-1, 2) = (2 \cdot -1, 2 \cdot 2)
step 3
Adding the vectors: (3,−4)+(−2,4)=(3+(−2),−4+4)(3, -4) + (-2, 4) = (3 + (-2), -4 + 4)
step 4
Simplifying the components: (1,0)(1, 0)
step 5
The resulting vector is u+2v=(1,0)\mathbf{u}+2\mathbf{v} = (1, 0)
Answer
u+2v=(1,0)\mathbf{u}+2\mathbf{v} = (1, 0)
Key Concept
Vector Addition and Scalar Multiplication
Explanation
To add a scalar multiple of a vector to another vector, first multiply the vector by the scalar, then add the corresponding components.
Solution by Steps
step 1
To find 2v−u2\mathbf{v}-\mathbf{u}, we first multiply v\mathbf{v} by 2 and then subtract u\mathbf{u} from the result
step 2
Multiplying v\mathbf{v} by 2: 2(−1,2)=(2⋅−1,2⋅2)2(-1, 2) = (2 \cdot -1, 2 \cdot 2)
step 3
Subtracting u\mathbf{u} from 2v2\mathbf{v}: (−2,4)−(3,−4)=(−2−3,4−(−4))(-2, 4) - (3, -4) = (-2 - 3, 4 - (-4))
step 4
Simplifying the components: (−5,8)(-5, 8)
step 5
The resulting vector is 2v−u=(−5,8)2\mathbf{v}-\mathbf{u} = (-5, 8)
Answer
2v−u=(−5,8)2\mathbf{v}-\mathbf{u} = (-5, 8)
Key Concept
Vector Subtraction and Scalar Multiplication
Explanation
To subtract a vector from a scalar multiple of another vector, first multiply the vector by the scalar, then subtract the corresponding components.
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