Problem 70
Question
How do you determine if two vectors are orthogonal?
Step-by-Step Solution
Verified Answer
Two vectors are orthogonal if their dot product is zero. If the dot product of two vectors is not zero, they are not orthogonal.
1Step 1: Understand Vector Orthogonality
Vector orthogonality or perpendicularity in higher dimensions works analogously to how one understands angles in two dimensions. Two vectors are orthogonal if their dot product is equal to zero.
2Step 2: Defining Dot Product
The Dot Product (or Scalar Product) of two vectors is a number (scalar) that is obtained by performing a specific operation on the coordinate vectors. For two vectors \( \mathbf{a} = [a_1, a_2] \) and \( \mathbf{b} = [b_1, b_2] \), the dot product is defined as \( \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \).
3Step 3: Check for Orthogonality
If \( \mathbf{a} \cdot \mathbf{b} = 0 \), then the two vectors are orthogonal. Plug in the vector coordinates into the dot product formula and simplify. If the result is zero, the vectors are orthogonal. If not, they are not orthogonal.
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