– Vectors in 2D and 3D
Vectors in 2D are \(\vec{u}\) = \(\dbinom{x\vec{i}}{y\vec{j}}\) or \(\vec{u}\) = \(x\vec{i} + y\vec{j}\)
Vectors in 3D are \(\vec{y}\) = \(x\vec{i} + y\vec{j} + z\vec{k}\) or
– Magnitude and Direction of a vector
Magnitude of the vector \(\vec{u}\) , |\(\vec{u}\)| =\(\sqrt{x^2 + y^2}\) and the direction is θ = \(tan^{-1}\dbinom{y}{x}\)
Unit vectors are \(\hat{i}\) = \(\dbinom{1}{0}\) and \(\hat{j}\) = \(\dbinom{1}{0}\)
– Properties of vectors
Addition and Subtraction of vectors \(\vec{u}\) = \(\dbinom{x_1}{y_1}\)
\(\vec{v}\) = \(\dbinom{x_2}{y_2}\) is
\(\vec{w}\) = \(\vec{u}\) ± \(\vec{v}\) = \(\dbinom{x_1 ± x_2 }{y_1 ± y_2}\)
Chasles’ rule
\(\vec{AB}+ \vec{BC}\) = \(\vec{AC}\)
– Use of vectors
The distance between 2 points (x1 , y1 , z1) and x2 , y2 , z2
d2 = (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
– Solving problems with vectors
Problems related to position vectors, velocity, acceleration , forces, work and moments.
Work (scalar) is a dot product of 2 vectors,
\(W\) = \(\vec{F}+ \vec{D}\) = |\(\vec{F}\)| × |\(\vec{d}\)| × cosθ
Moment (vector) is a cross product
\(M\) = \(\vec{F}+ \vec{d}\) = \(\vec{k}\) (\(F_x × d_y - F_y × d_x\)
The moment is perpendicular to the plane containing the 2 vectors.
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