Math Problem Statement

Test:MAC1114-T4 Question 6 of 20 This test: 20 point(s) possible This question: 1 point(s) possible

Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Question 16 Question 17 Question 18 Question 19 Question 20 Question content area top left Part 1 Use the vectors in the figure at the right to graph the following vector. 3 Bold v plus Bold u minus 2 Bold w . . . Question content area top right Part 1 Bold vBold wBold u

A coordinate system has a horizontal axis marked with 10 equally spaced tick marks and a vertical axis marked with 10 equally spaced tick marks. The following three vectors are shown: u, from (5, 5) to (9, 3); v, from (2, 2) to (2, 5); w, from (3, 5) to (4, 7). Question content area bottom Part 1 Select the figure with a vector that is equivalent to 3 Bold v plus Bold u minus 2 Bold w. A.

A coordinate system has a horizontal axis marked with 10 equally spaced tick marks and a vertical axis marked with 17 equally spaced tick marks. A vector extends from (2, 3) to (8, 14). B.

A coordinate system has a horizontal axis marked with 10 equally spaced tick marks and a vertical axis marked with 17 equally spaced tick marks. A vector extends from (8, 5) to (2, 12). C.

A coordinate system has a horizontal axis marked with 10 equally spaced tick marks and a vertical axis marked with 17 equally spaced tick marks. A vector extends from (6, 1) to (4, 16). D.

A coordinate system has a horizontal axis marked with 10 equally spaced tick marks and a vertical axis marked with 17 equally spaced tick marks. A vector extends from (4, 7) to (6, 10). Select the figure with a vector that is equivalent to 3 Bold v plus Bold u minus 2 Bold w. Time Remaining: 02:24:40

Solution

To solve the problem, we need to calculate the resulting vector 3v+u−2w3\mathbf{v} + \mathbf{u} - 2\mathbf{w}. Here's the step-by-step process:

1. Break Down Each Vector:

  • v\mathbf{v}: Starts at (2,2)(2, 2) and ends at (2,5)(2, 5), so: v=(2−2,5−2)=(0,3)\mathbf{v} = (2 - 2, 5 - 2) = (0, 3)
  • u\mathbf{u}: Starts at (5,5)(5, 5) and ends at (9,3)(9, 3), so: u=(9−5,3−5)=(4,−2)\mathbf{u} = (9 - 5, 3 - 5) = (4, -2)
  • w\mathbf{w}: Starts at (3,5)(3, 5) and ends at (4,7)(4, 7), so: w=(4−3,7−5)=(1,2)\mathbf{w} = (4 - 3, 7 - 5) = (1, 2)

2. Scale and Combine the Vectors:

  • 3v3\mathbf{v}: Multiply (0,3)(0, 3) by 3: 3v=(0,9)3\mathbf{v} = (0, 9)
  • −2w-2\mathbf{w}: Multiply (1,2)(1, 2) by −2-2: −2w=(−2,−4)-2\mathbf{w} = (-2, -4)
  • Sum of all components: Add 3v3\mathbf{v}, u\mathbf{u}, and −2w-2\mathbf{w}: (0,9)+(4,−2)+(−2,−4)=(0+4−2,9−2−4)=(2,3)(0, 9) + (4, -2) + (-2, -4) = (0 + 4 - 2, 9 - 2 - 4) = (2, 3)

3. Resulting Vector:

The resulting vector starts at the origin (0,0)(0, 0) and ends at (2,3)(2, 3). Based on the provided answer choices, select the figure that matches this vector.

Final Answer:

Inspect the given figures to match the vector coordinates. From your description:

  • Answer: Figure D.

Do you want a breakdown of why the other options don't fit?

Related Questions:

  1. What are the properties of vector addition and scalar multiplication?
  2. How do you find the magnitude of a vector?
  3. What role does direction play in vector problems like this?
  4. How do transformations like reflection or rotation affect vector calculations?
  5. How can we verify the correctness of vector operations graphically?

Tip:

Always recheck your calculations for scaled vectors before summing them to avoid simple arithmetic errors.

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Math Problem Analysis

Mathematical Concepts

Vector Addition
Scalar Multiplication
Coordinate Geometry

Formulas

Vector Addition: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
Scalar Multiplication: k(x, y) = (kx, ky)

Theorems

Properties of Vectors

Suitable Grade Level

College-level (Calculus and Linear Algebra)

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