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MATH1002 Linear Algebra
创建日期：2024-06-01 15:38:05
所属分类：数学mathematics
标签： MATH1002
Linear Algebra

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MATH1002 Linear Algebra

Answers are provided below.
Important Ideas and Useful Facts:
(i) In R3, the cross product of u = [u1, u2, u3] and v = [v1, v2, v3] is the vector
u× v = [u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1].
(ii) The cross product u× v is orthogonal to both u and v.
(iii) The direction of u× v is given by the right-hand rule.
(iv) Anti-commutativity of cross product: For all vectors u and v in R3, u× v = −(v × u).
(v) Distributivity of cross product: For all vectors u, v and w in R3,
(u+ v)×w = u×w + v ×w.
(vi) If v and w are vectors in R3 and c is a scalar then (cv)×w = c(v ×w) = v × (cw).
(vii) For all vectors u in R3, u× u = 0.
(viii) If θ is the angle between vectors v and w in R3 then
‖v ×w‖ = ‖v‖‖w‖ sin θ.
(ix) The area of the parallelogram inscribed by v and w in R3 is ‖v × w‖, and the area of the triangle
inscribed by v and w is 1
2
‖v ×w‖.
(x) A normal vector for a line ℓ is a nonzero vector n which is orthogonal to any vector parallel to ℓ.
(xi) A direction vector for a line ℓ is a nonzero vector d which is parallel to ℓ. If A and B are distinct
points on ℓ then
−−→
AB is a direction vector for ℓ.
(xii) The normal form of the equation for a line ℓ in R2 is
n · (x− p) = 0, or, equivalently, n · x = n · p,
where p is a point on ℓ, and n is a normal vector for ℓ.
(xiii) The general form of the equation for a line ℓ in R2 is
ax+ by = c,
where n = [a, b] is a normal vector for ℓ.
(xiv) The vector form of the equation for a line ℓ in R2 or R3 is
x = p+ td,
where p is a point on ℓ, d is a direction vector for ℓ and t ∈ R.
(xv) The parametric equations of ℓ are the equations corresponding to the components:
x = p1 + td1 and y = p2 + td2 (and, for lines in R
3, z = p3 + td3), for t ∈ R.
1
Preparatory Exercises:
1. Given that a = [2,−1, 2],b = [1, 1,−1] and c = [3, 0,−4], find
(i) a× b
(ii) the angle between a and a× b
(iii) ‖a× b‖
(iv) the sine of the angle between a and b.
2. Let ℓ be the line in R3 passing through the point (2, 3, 5) in the direction of [1, 3,−1]. Find a vector
form for ℓ and parametric equations for ℓ.
Answers:
1. (i) [−1, 4, 3] (ii) pi
2
(iii)
√
26 (iv)
√
78
9
2. x = [2, 3, 5] + t[1, 3,−1] where t ∈ R,
x = 2 + t
y = 3 + 3t
z = 5− t

 t ∈ R.
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