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1. Mohd Arif Fahmi bin Rosli
Lecture 1: Introduction & Vector Function of One Variable

2.  Scalar and Vector
 If r = axi + ayj + azk, then modulus of |r| is…
 Direction cosines [l, m, n] are the cosines of the angles between vector r and the
axes:
 Let r = axi + ayj + azk, then
 l = cos α =
ax
|r|
; m = cos β =
ay
|r|
; n = cos γ =
az
|r|
 l2 + m2 + n2 = 1
 Scalar product / dot product
 A · B = AB cos θ = |A||B| cos θ = axbx + ayby + azbz = B · A

3.  Vector product / cross product
 |A x B| = AB sin θ
 A x B = = (aybz – azby)i – (axbz – azbx)j + (axbz – azbx)k = -(B x A)
 Angle between two vectors
 cos θ = l1l2 + m1m2 + n1n2
 Where l1, m1, n1 and l2, m2, n2 are the direction cosines of vector
r1 and r2 respectively.

4.  If i, j, k are unit vectors in the directions OX, OY, OZ, respectively. Vector r is in
direction of OP and P is the point (3, 2, 6). Determines direction cosines of the
vector r, (i.e. l, m, n)

5. Determine the angle between
the vectors A and B, θ

6.  A · (B x C)
 A · (B x C) = =
 Properties of triple products
1. A · (B x C) = C · (A x B) = B · (C x A) CYCLIC ORDER
2. B · (A x C) ≠ A · (B x C) = -A · (B x C) NOT IN CYCLIC ORDER
3. A · (B x A) = B · (A x B) = C · (A x C) = 0 TWO VECS ARE IDENTICAL

7. Determination of Coplanar
 The magnitude of scalar triple product |A · (B x C)| is equal to the volume of the
parallelepiped.
A · (B x C) = A · (BC sin θ n) = ABC sin θ cos ϕ, where θ is the angle between B
and C and ϕ is the angle between A and n.
 Therefore, |A · (B x C)|= ABC |sin θ cos ϕ|
 BC |sin θ| is the parallelogram area
 A|cos ϕ| is the altitude/ height of the parallelepiped.

8.  Consequently, if A · (B x C) = 0, then the volume of the parallelepiped is zero
and those vectors are coplanar

9. A x (B x C)
 Then (B x C) is a vector perpendicular to the plane of B and C and A x (B x
C) is a vector perpendicular to the plane containing A and (B x C), i.e.
coplanar with B and C.
 Thus,

10.  Also,
 Proof can be seen on Appendix K.A. Stroud, Dexter J. Booth.(2007). Advanced Engineering
Mathematics, 6th Edition.

11.  If A is a vector and depends on the scalar variable t, time, then A can be
represented as A(t) and A is said to be a function of t.
 Differentiating A(t) with respect to t, gives..
 Not only with respect to t, differentiation also can be done with respect to other
such u.

12.  Let’s visualise
 If a position vector OP moves to OQ when u becomes u + δu,
then as δu → 0, the direction of the chord PQ becomes that of
the tangent to the curve at P
 i.e. the direction of
d𝐀
d𝑢
is along the tangent to the locus of P.
 Details can be found in K.A. Stroud, Dexter J. Booth (2013)
Engineering Mathematics, 7th Edition. Palgrave Macmillan. (page 322)

13.  Let r(t) = (cos t, sint, t)
1. Sketch and describe the curve associated with r
2. If particle travels along this curve (with t representing time) then calculate r’(t)
and show that its speed is constant