Ch 10 introduces vector algebra in 3D. Students learn about types of vectors, addition, scalar multiplication, dot (scalar) product, cross (vector) product, and the scalar triple product.
Vectors have magnitude and direction. Types: zero, unit, collinear, coplanar. Position vector of P(x,y,z) = xi + yj + zk. Magnitude |a⃗| = √(a₁²+a₂²+a₃²). Unit vector â = a⃗/|a⃗|.
Dot product: a⃗·b⃗ = |a||b|cos θ = a₁b₁+a₂b₂+a₃b₃. Cross product: a⃗×b⃗ = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|, perpendicular to both, |a⃗×b⃗| = |a||b|sin θ.
[a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗) = volume of parallelepiped. If [a⃗ b⃗ c⃗] = 0, vectors are coplanar. [a⃗ b⃗ c⃗] = [b⃗ c⃗ a⃗] = [c⃗ a⃗ b⃗] (cyclic).
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The cross product a⃗×b⃗ gives a vector perpendicular to both a⃗ and b⃗. Its magnitude |a⃗×b⃗| = |a||b|sin θ equals the area of the parallelogram formed by the two vectors. Direction follows the right-hand rule.
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