Calculus Volume 3 · 免费预览
内积
内积将对应分量相乘后求和；当结果为零时可判定正交性，通过余弦公式度量夹角，并可将一个向量投影到另一个向量上，以便计算功和进行分解。
1. Definition and properties
Definition and properties
2. The angle formula
The angle formula
3. Projections
Projections
4. Direction cosines and the Cauchy-Schwarz inequality
Direction cosines and the Cauchy-Schwarz inequality
5. Dot product value
The dot product of (1, 2, 3) and (4, -1, 2) equals what number?
6. Orthogonality test
The vectors (1, 2, -1) and (2, -1, 0) are which relation?
7. Angle formula
The cosine of the angle between u and v equals which expression?
8. Projection formula
The projection of u onto v equals which vector?
9. Why dot product is a scalar
The dot product differs from vector addition in producing what kind of mathematical object?
10. Self dot product
The dot product of v with itself equals the square of which quantity? Answer with one word.
11. Work formula
Work equals force dot displacement. As a two-word phrase, what is it the product of? Answer like vector product.
12. Cauchy-Schwarz direction
The Cauchy-Schwarz inequality says |u dot v| is at most what product? Answer like the product of the magnitudes, in words.
13. Scalar projection versus vector projection
The scalar projection of u onto v is u dot v over |v|, while the vector projection additionally does what?
14. Direction cosines sum
The squares of the three direction cosines sum to what number? Answer with a number.