Math Problem Statement
Let u = (1, 2, 3),
v = (2, 2, −1), and w = (4, −3, −1). Find z, where 2u + v − w + 3z = 0.
Solution
We are given the equation:
where , , and . We need to find .
Step 1: Expand the terms
First, let's compute each vector component separately.
Now substitute these values into the equation:
Step 2: Simplify the left-hand side
Start by adding the vectors , , and :
So the equation becomes:
Step 3: Solve for
Now, we want to isolate . Subtract from both sides:
Finally, divide both sides by 3 to find :
Final Answer:
Would you like more details or have any questions about this solution? Here's some follow-up:
- How to compute the magnitude of a vector?
- What is the dot product of and ?
- How to check if two vectors are perpendicular?
- What is the cross product of and ?
- How to find the angle between two vectors?
Tip: When working with vector equations, break down each component separately, and remember to handle scalar multiplications and vector additions/subtractions one component at a time for accuracy.
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Linear Equations
Vector Addition
Scalar Multiplication
Formulas
Vector addition: (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3)
Scalar multiplication: c(a1, a2, a3) = (ca1, ca2, ca3)
Solving for unknown vector: Ax + B = 0 ⟹ x = -B/A
Theorems
Properties of Vector Addition and Scalar Multiplication
Suitable Grade Level
Grades 10-12