Math Problem Statement

Given vectors a = (1, 2, 0) and b = (m, 5, 6) with a ⊥ b, find the value of m.

Solution

The problem states that we are given the vectors:

a=(1;2;0),b=(m;5;6)\vec{a} = (1; 2; 0), \quad \vec{b} = (m; 5; 6) and the condition that ab\vec{a} \perp \vec{b}, which means the vectors a\vec{a} and b\vec{b} are orthogonal. The task is to find the value of mm such that ab=0\vec{a} \cdot \vec{b} = 0.

The dot product of two vectors a=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3) and b=(b1,b2,b3)\vec{b} = (b_1, b_2, b_3) is given by:

ab=a1b1+a2b2+a3b3\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3

For the given vectors, we compute:

ab=1m+25+06\vec{a} \cdot \vec{b} = 1 \cdot m + 2 \cdot 5 + 0 \cdot 6 ab=m+10\vec{a} \cdot \vec{b} = m + 10

Since ab\vec{a} \perp \vec{b}, the dot product must be zero:

m+10=0m + 10 = 0

Solving for mm, we get:

m=10m = -10

Thus, the value of mm is 10-10.

Would you like more details or have any questions?

Here are 5 related questions to expand this topic:

  1. What is the geometric interpretation of two vectors being orthogonal?
  2. How do we calculate the magnitude of a vector?
  3. What happens if vectors are not orthogonal but have a non-zero dot product?
  4. Can two non-zero vectors be orthogonal in a higher-dimensional space?
  5. 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