Math Problem Statement
Let u = 3i − k, let v = 2i + 3j − 4k, and let w = −i + 5j − 3k. a) Compute u + 2j − k. b) Write the answer to part (a) in coordinates (i.e. without using i, j, knotation). c) Compute 2u − 3w. d) If x is a vector such that 1 3 x + u = v + w, find x
Solution
Let's tackle each part of the problem step by step.
Given:
a) Compute
First, express in component form:
Next, add to . In component form:
Now, perform the vector addition:
b) Write the answer to part (a) in coordinates (i.e., without using notation)
The answer from part (a) is already in coordinate form:
c) Compute
First, find and :
Now, subtract from :
d) If is a vector such that , find
Start by calculating :
= (2 - 1, 3 + 5, -4 - 3) = (1, 8, -7)$$ Given: $$\frac{1}{3} \mathbf{x} + \mathbf{u} = (1, 8, -7)$$ We know $$\mathbf{u} = (3, 0, -1)$$, so: $$\frac{1}{3} \mathbf{x} + (3, 0, -1) = (1, 8, -7)$$ Subtract $$\mathbf{u}$$ from both sides: $$\frac{1}{3} \mathbf{x} = (1, 8, -7) - (3, 0, -1) = (1 - 3, 8 - 0, -7 + 1) = (-2, 8, -6)$$ Multiply by 3 to solve for $$\mathbf{x}$$: $$\mathbf{x} = 3(-2, 8, -6) = (-6, 24, -18)$$ --- **Summary of Results:** - a) $$\mathbf{u} + 2\mathbf{j} - \mathbf{k} = (3, 2, -2)$$ - b) $$(3, 2, -2)$$ - c) $$2\mathbf{u} - 3\mathbf{w} = (9, -15, 7)$$ - d) $$\mathbf{x} = (-6, 24, -18)$$ Do you have any further questions or need more details? ### Relative Questions: 1. How do you compute the magnitude of a vector like $$\mathbf{x}$$ from part (d)? 2. What is the geometric interpretation of adding and subtracting vectors? 3. Can you solve for a vector $$\mathbf{y}$$ if given an equation like $$\mathbf{y} + 2\mathbf{z} = \mathbf{a} - \mathbf{b}$$? 4. What happens if you multiply a vector by a negative scalar, such as $$-2\mathbf{u}$$? 5. How would you find a unit vector in the direction of $$\mathbf{u}$$? ### Tip: Always check your arithmetic carefully when adding or subtracting vectors, especially when working with multiple operations like scalar multiplication and addition.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vectors
Vector Addition
Scalar Multiplication
Formulas
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Theorems
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Suitable Grade Level
Undergraduate