Chapter 11

Vectors in Three Dimensions (Pure 3)

Vectors in space, the scalar product and the angle it measures, the vector equation of a line, and deciding whether two lines meet, are parallel, or are skew.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Measurement and Geometry)ตรงกับ ม.5 เวกเตอร์ในสามมิติ (เพิ่มเติม) — ส่วนสมการเวกเตอร์ของเส้นตรงและเส้นตรงไขว้ไม่มีในหลักสูตรไทย
Common Core (US): N-VM
Cambridge: 9709 Pure Mathematics 3 — Vectors

11.1 The scalar product

Two ways of writing the same number

a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} = ∣ a ∣ ∣ b ∣ cos θ

The left-hand version is arithmetic you can always do; the right-hand version is what it MEANS. Setting them equal is how an angle is found.

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What the sign of the scalar product tells you
- zero — the vectors are perpendicular
- positive — the angle between them is acute
- negative — the angle is obtuse

ตัวอย่าง

Find the acute angle between $a = 2 i - j + 2 k$ and $b = 3 i + 4 k$ .

วิธีทำ

1. $a \cdot b = 6 + 0 + 8 = 14$
2. $∣ a ∣ = 3$ and $∣ b ∣ = 5$
3. $cos θ = \frac{14}{15}$

ตอบ $θ \approx 21. 0^{\circ}$

ข้อควรระวัง

A scalar product is a NUMBER, not a vector. An answer written as a column of three entries means the multiplication was done component by component without adding them up, which is a different operation entirely.

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