Vectors
Addition of two vectors
𝗔 = 15∠10º 𝐁 = 8∠30º
And you want 𝐑 = vector sum of 𝗔 + 𝐁

Break each vector into x and y components.
Ax = 15 cos 10 = 14.8
Ay = 15 sin 10 = 2.60
Bx = 8 cos 30 = 6.98
By = 8 sin 30 = 4.00

Add the two x & y components to get the x/y component of the resultant, R
Rx = Ax + Bx = 21.8
Ry = Ay + By = 6.6

Now convert resultant into magnitude and angle
R = √(Rx² + Ry²) = √(475 + 43.6) = 22.8
θ = arctan (Ry/Rx) = arctan (21.8/6.6) = 16.8º

Alternate Method for magnitude
A = 15∠10º B = 8∠30º
R² = A² + B² + 2ABcosθ
R² = 15² + 8² + 2•15•8cos20 = 515
R = 22.7

Cosine rule
As applied to two vectors, 𝗔 and 𝐁 with angle between them of θ, the resultant 𝐑 of the two is:
𝐑 = 𝗔 + 𝐁
R² = A² + B² + 2ABcosθ
𝐑 = 𝗔 – 𝐁
R² = A² + B² – 2ABcosθ

Cross Product
is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them. If either of the vectors being multiplied is zero or the vectors are parallel then their cross product is zero.

The cross product 𝐚×𝐛 is defined as a vector 𝐜 that is perpendicular to both 𝐚 and 𝐛, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
𝐜 = 𝐚×𝐛 = ab𝐧 sinθ
θ is the angle between 𝐚 and 𝐛
𝐧 is the unit vector in the direction defined by the right hand rule.
𝐚×𝐛 = –𝐛x𝐚

If 𝐚 = a₁𝐢 + a₂𝐣 + a₃𝐤
If 𝐛 = b₁𝐢 + b₂𝐣 + b₃𝐤
𝐚×𝐛 = (a₂b₃–a₃b₂)𝐢 + (a₃b₁–a₁b₃)𝐣 + (a₁b₂–a₂b₁)𝐤

Dot Product
The dot product of two vectors results in a scalar.
Two vectors 𝐚 and 𝐛 with magnitudes a and b:
𝐚·𝐛 = ab cosθ

In two-dimensional space, the dot product of
vectors [a, b] and [c, d] is ac + bd.
In a three-dimensional space, the dot product of
vectors [a, b, c] and [d, e, f] is ad + be + cf.
two vectors are perpendicular if the dot product is zero
ac + bd = 0 (2D)