Math Problem Statement
Solution
The problem states that we are given the vectors:
and the condition that , which means the vectors and are orthogonal. The task is to find the value of such that .
The dot product of two vectors and is given by:
For the given vectors, we compute:
Since , the dot product must be zero:
Solving for , we get:
Thus, the value of is .
Would you like more details or have any questions?
Here are 5 related questions to expand this topic:
- What is the geometric interpretation of two vectors being orthogonal?
- How do we calculate the magnitude of a vector?
- What happens if vectors are not orthogonal but have a non-zero dot product?
- Can two non-zero vectors be orthogonal in a higher-dimensional space?
- How can the cross product help in finding a vector perpendicular to two given vectors?
Tip: When calculating the dot product of two vectors, always check if each component contributes correctly to avoid mistakes in simplifications.
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Math Problem Analysis
Mathematical Concepts
Vectors
Dot Product
Orthogonal Vectors
Formulas
Dot product: a · b = a1*b1 + a2*b2 + a3*b3
Theorems
Orthogonality: Two vectors are orthogonal if their dot product is zero.
Suitable Grade Level
Grades 9-11
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