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) Home Area, Volume Atomic Mass Black Body Radiation Boolean Algebra Calculus Capacitor Center of Mass Carnot Cycle Charge Chemistry   Elements   Reactions Circuits Complex numbers Constants Curves, lines deciBell Density Electronics Elements Flow in fluids Fourier's Law Gases Gravitation Greek Alphabet Horizon Distance Interest Magnetics Math   Trig Math, complex Maxwell's Eq's Motion Newton's Laws Octal/Hex Codes Orbital Mechanics Particles Parts, Analog IC   Digital IC   Discrete Pendulum Planets Pressure Prime Numbers Questions Radiation Refraction Relativistic Motion Resistance, Resistivity Rotation Series SI (metric) prefixes Skin Effect Specific Heat Springs Stellar magnitude Thermal Thermal Conductivity Thermal Expansion Thermodynamics Trigonometry Units, Conversions Vectors Volume, Area Water Wave Motion Wire, Cu   Al   metric Young's Modulus