Vectors

How to calculate direction cosine in i direction

a = (i, j , k)

1) Modulus = √(i)² + (j)² + (k)²

2) i / modulus

Scalar Multiplication / Dot product

a.b = a₁b₁ + a₂b₂ + a₃b₃

ADD TOGETHER

Scalar Multiplication in geometric form

a.b = |a|x|b|xcos(θ)

2 forms of distributive law

a . (λb) = (aλ).b = λ(a.b)

a.(b+c) = a.b + a.c

How to find unit vector perpendicular to two vectors

1) a x b

2) modulus

3) (1/modulus) (a x b)

How to prove 3 vectors are coplanar

If scalar triplet = 0, then coplanar.

1) b x c

2) a . [b x c]

Vector Product

a = a₁, a₂ , a₃

b = b₁, b₂ , b₃

a x b = ((a₂b₃) - (a₃b₂)) , - ((a₁b₃) - (a₃b₁)) , ((a₁b₂) - (a₂b₁))

DONT ADD TOGETHER

Vector Product in geometric form

a x b = |a| x |b| x sinθ

How to find component of a force ( F = a, b ,c) in direction (x,y,z) ABOUT ORIGIN

1) r = (x, y, z)

2) F . r

3) Modulus r

4) component = (F.r) / modulus r

Moment of a force

M = r x F

or

M = |r| x |F| x sinθ

Moment of a force about a point

Subtract original r from point

ie/ (r₁ r₂ r₃) - (p₁ p₂ p₃)

Area of a triangle

1) XY

2) XZ

3) XY x XZ

4) Modulus of this

5) divide by two

Equation of a straight line between two points (a & b)

1) find line ab (-OA + OB)

2) r = a + t(b-a)

sub in the values of the vectors for a and b-a