Math Problem Statement

Let Bold u equals negative 7 Bold i plus 8 Bold j ​, Bold v equals 8 Bold i minus Bold j ​, and Bold w equals negative 2 Bold i . Find 4 Bold u minus left parenthesis 3 Bold v minus Bold w right parenthesis . Question content area bottom Part 1 4 Bold u minus left parenthesis 3 Bold v minus Bold w right parenthesis

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Solution

We are given the following vectors:

u=7i+8j,v=8ij,w=2i\mathbf{u} = -7 \mathbf{i} + 8 \mathbf{j}, \quad \mathbf{v} = 8 \mathbf{i} - \mathbf{j}, \quad \mathbf{w} = -2 \mathbf{i}

We need to compute:

4u(3vw)4 \mathbf{u} - (3 \mathbf{v} - \mathbf{w})

Step 1: Compute 4u4 \mathbf{u}

Multiply each component of vector u\mathbf{u} by 4:

4u=4(7i+8j)=28i+32j4 \mathbf{u} = 4 \left( -7 \mathbf{i} + 8 \mathbf{j} \right) = -28 \mathbf{i} + 32 \mathbf{j}

Step 2: Compute 3v3 \mathbf{v}

Multiply each component of vector v\mathbf{v} by 3:

3v=3(8ij)=24i3j3 \mathbf{v} = 3 \left( 8 \mathbf{i} - \mathbf{j} \right) = 24 \mathbf{i} - 3 \mathbf{j}

Step 3: Compute 3vw3 \mathbf{v} - \mathbf{w}

Now subtract vector w\mathbf{w} from 3v3 \mathbf{v}:

3vw=(24i3j)(2i)=24i3j+2i3 \mathbf{v} - \mathbf{w} = (24 \mathbf{i} - 3 \mathbf{j}) - (-2 \mathbf{i}) = 24 \mathbf{i} - 3 \mathbf{j} + 2 \mathbf{i}

Simplifying:

3vw=(24+2)i3j=26i3j3 \mathbf{v} - \mathbf{w} = (24 + 2) \mathbf{i} - 3 \mathbf{j} = 26 \mathbf{i} - 3 \mathbf{j}

Step 4: Compute 4u(3vw)4 \mathbf{u} - (3 \mathbf{v} - \mathbf{w})

Now subtract (3vw)(3 \mathbf{v} - \mathbf{w}) from 4u4 \mathbf{u}:

4u(3vw)=(28i+32j)(26i3j)4 \mathbf{u} - (3 \mathbf{v} - \mathbf{w}) = (-28 \mathbf{i} + 32 \mathbf{j}) - (26 \mathbf{i} - 3 \mathbf{j})

Distribute the subtraction:

=(28i+32j)26i+3j= (-28 \mathbf{i} + 32 \mathbf{j}) - 26 \mathbf{i} + 3 \mathbf{j}

Simplifying:

=(2826)i+(32+3)j= (-28 - 26) \mathbf{i} + (32 + 3) \mathbf{j}

=54i+35j= -54 \mathbf{i} + 35 \mathbf{j}

Final Answer:

4u(3vw)=54i+35j4 \mathbf{u} - (3 \mathbf{v} - \mathbf{w}) = -54 \mathbf{i} + 35 \mathbf{j}


Would you like to see more details or explore any specific steps? Here are a few related questions:

  1. How would you compute a similar expression involving 5 vectors?
  2. What is the geometric interpretation of subtracting vectors?
  3. How do you find the magnitude of a vector resulting from an operation like this?
  4. How do you apply vector operations to solve real-world problems?
  5. How would you extend this to 3-dimensional vectors?

Tip: When performing vector operations, always break each vector into its components and deal with them separately.

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Math Problem Analysis

Mathematical Concepts

Vector Operations
Vector Addition and Subtraction
Scalar Multiplication

Formulas

Scalar Multiplication: c * (ai + bj) = (c * a)i + (c * b)j
Vector Addition/Subtraction: (ai + bj) ± (ci + dj) = (a ± c)i + (b ± d)j

Theorems

Vector Addition is Commutative and Associative

Suitable Grade Level

Grades 10-12