Math Problem Statement
Given v⃗ v→ with an initial point of ⟨−4,−2,1⟩⟨−4,−2,1⟩ and a terminal point of ⟨3,6,−6⟩⟨3,6,−6⟩ , find a unit vector in the same direction as v⃗ v→.
Solution
To find a unit vector in the same direction as with an initial point and a terminal point , we need to follow these steps:
- Find the vector from the initial and terminal points.
- Calculate the magnitude of .
- Divide the vector by its magnitude to get the unit vector.
Step 1: Find the vector
The vector is given by the difference between the terminal point and the initial point:
Step 2: Calculate the magnitude of
The magnitude of , denoted as , is given by:
Step 3: Divide the vector by its magnitude to get the unit vector
The unit vector in the direction of is given by:
To rationalize the denominator:
Thus, the unit vector in the same direction as is:
Would you like any more details or have any questions about this solution?
Here are some related questions:
- How do you find the direction cosines of a vector?
- What is the difference between a unit vector and a normal vector?
- How can unit vectors be used in physics?
- What is the significance of the dot product of unit vectors?
- How do you normalize a vector in higher dimensions?
- What is the role of unit vectors in coordinate systems?
- Can you explain the concept of vector projection?
- How do you calculate the angle between two vectors using their unit vectors?
Tip: Unit vectors are especially useful in defining directions and are often used to simplify vector calculations in physics and engineering.
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Math Problem Analysis
Mathematical Concepts
Vectors
Vector Operations
Unit Vectors
Formulas
Vector subtraction
Magnitude of a vector
Unit vector formula
Theorems
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Suitable Grade Level
High School
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