Vectors

Let \(P=(1,2)\) and \(Q=(5,4)\), write \(\overrightarrow{PQ}\) in component form and find the magnitude of the vector.

Let \(\mathbf{u} = 4 \vec{i} + 6 \vec{j}\) and \(\mathbf{v} = 2 \vec{i} - 4 \vec{j}.\)

1. \(\mathbf{u} + \mathbf{v}\)
2. \(\dfrac{3}{2} \mathbf{u}\)
3. \(\mathbf{u} - \mathbf{v}\)

Find the dot product of \(\mathbf{u}\) and  \(\mathbf{v}\) for the following vectors:

1. \(\mathbf{u}= \langle 0,0 \rangle,\) \(\mathbf{v}= \langle 4,2 \rangle\)
2. \(\mathbf{u}=\langle 1,1 \rangle ,\) \(\mathbf{v}= \langle 6,10 \rangle\)
3. \(\mathbf{u}=\langle 12, 8 \rangle ,\) \(\mathbf{v}= \langle 2, -3 \rangle\)

Let \(\mathbf{v}\) be a vector with initial point \(P=(2,5)\) and terminal point \(Q=(5,13)\). Find a unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}.\)