In space as well, the dot product is →a·→b = a₁b₁ + a₂b₂ + a₃b₃ = |→a||→b|cosθ
Two vectors are orthogonal ⇔ →a·→b = 0
Example ① Find the angle θ between →a=(1,2,3) and →b=(2,−1,1).
Solution: →a·→b = 2−2+3 = 3。|→a|=√14、|→b|=√6。
cosθ = 3/(√14·√6) = 3/√84 = √21/14
Solution: →a·→b = 2−2+3 = 3。|→a|=√14、|→b|=√6。
cosθ = 3/(√14·√6) = 3/√84 = √21/14
Key Points
- The dot-product formula is the same as in a plane: the sum of products of corresponding components
- Orthogonality: dot product = 0
Practice Problems
- Find k such that →a=(2,1,k) and →b=(2,−1,3) are orthogonal.
Answers and Explanations
- Answer: k = −1
4−1+3k=0 → k=−1。