Calculus Volume 3 · free preview
The Dot Product
The dot product multiplies corresponding components and sums, detecting orthogonality when zero, measuring angles through the cosine formula, and projecting one vector onto another for work and decomposition.
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.
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