2 vectors
a vector can be expressed as \(\vec{AB}\) or \(\tilde{u}\)
\[\tilde{u}=u_1\tilde{i}+u_2\tilde{j}+u_3\tilde{k}\]
2.1 magnitude and unit vectors
the magnitude of a vector \(|\tilde{u}|\) is the distance of that vector \(\tilde{u}\)
\[|\tilde{u}|=\sqrt{u_1^2+u_2^2+u_3^2}\]
a unit vector \(\hat{u}\) is any vector with magnitude 1
\[\hat{u}=\frac{\tilde{u}}{|\tilde{u}|}\]
2.2 addition and subtraction
\[\tilde{u}+\tilde{v}=(u_1+v_1)\tilde{i}+(u_2+v_2)\tilde{j}+(u_3+v_3)\tilde{k}\]
\[\tilde{u}-\tilde{v}=(u_1-v_1)\tilde{i}+(u_2-v_2)\tilde{j}+(u_3-v_3)\tilde{k}\]
2.3 the zero vector
the zero vector \(\tilde{0}\) has magnitude \(0\) and have no particular direction
\[\tilde{0}=0\tilde{i}+0\tilde{j}+0\tilde{k}\]
note that \(\tilde{u}-\tilde{u}\neq{}0\), but rather \(\tilde{u}-\tilde{u}=\tilde{0}\)
2.4 parallel vectors
two vectors \(\tilde{u}\) and \(\tilde{v}\) are parallel if \(\tilde{u}\) is a scalar multiple of \(\tilde{v}\)
\[\tilde{u}=k\tilde{v},k\in{R}\]
2.5 linear dependence
a set of vectors is linearly dependent if one can be expressed as a linear combination of the others
\[\tilde{u}=p\tilde{v}+q\tilde{w}\]
a set of vectors is linearly independent if its not linearly dependent
2.6 dot product
the dot product always return a real number which is the amount of space covered by the two vector
\[\tilde{u}\cdot{}\tilde{v}=|\tilde{u}||\tilde{v}|\cos(\theta)\] \[\tilde{u}\cdot\tilde{v}=u_1v_1+u_2v_2+u_3v_3\]
- if \(\tilde{u}\cdot\tilde{v}=|\tilde{u}||\tilde{v}|\) then \(\tilde{u}\) and \(\tilde{v}\) is parallel
- if \(\tilde{u}\cdot\tilde{v}=0\) and \(\tilde{u},\tilde{v}\neq\tilde{0}\) then \(\tilde{u}\) and \(\tilde{v}\) is perpendicular
2.7 vector resolutes
the scalar resolute of \(\tilde{u}\) in the direction of \(\tilde{v}\) is given by \(\tilde{u}\cdot\hat{v}\) \[\text{or...}\quad\frac{\tilde{u}\cdot\tilde{v}}{|\tilde{v}|}\]
the vector resolute of \(\tilde{u}\) in the direction of \(\tilde{v}\) is given by \((\tilde{u}\cdot\hat{v})\hat{v}\) \[\text{or...}\quad\frac{\tilde{u}\cdot\tilde{v}}{|\tilde{v}|}\tilde{v}\]
2.8 geometric proofs
2.8.1 triangles
equilateral triangles have equal length sides and equal angles of \(60^\circ\) at each vertex. it is sufficient to show only one of these properties
isosceles triangles have only two sides that are equal in length and two angles that are equal. it is sufficient to show only one of these properties
2.8.2 quadrilaterals
squares have equal angles of \(90^\circ\) at each vertex and equal side lengths. a sufficient proof requires showing that both of these properties are true
rectangles have equal angles of \(90^\circ\) at each vertex. a sufficient proof requires showing that both pairs of opposite sides are parallet and the there is one \(90^\circ\) angle
rhombuses have two pairs of opposite parallel sides, with all sides equal in length. a sufficient proof requires showing that both of these properties are true
parallelograms have two pairs of opposite parallel sides. a sufficient proof requires showing that a single pair of opposite sides is parallel and equal in length; the vectors representing two opposite sider are equal
trapeziums have one pair of parallel sides. a sufficient proof requires showing that a single pair of opposite sides is parallel