Vectors in two dimensions：练习题

Given the vectors \(\mathbf{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\), calculate the resultant vector \(\mathbf{a} + \mathbf{b}\).

Point \(A\) has coordinates \((2, 5)\) and point \(B\) has coordinates \((7, -3)\). Express the vector \(\vec{AB}\) as a column vector.

The vector \(\mathbf{m} = \begin{pmatrix} 4 \\ x \end{pmatrix}\) is parallel to the vector \(\mathbf{n} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\). Find the value of \(x\).

Calculate the magnitude of the vector \(\mathbf{v} = \begin{pmatrix} 5 \\ -12 \end{pmatrix}\).

The magnitude of the vector \(\mathbf{u} = \begin{pmatrix} k \\ 8 \end{pmatrix}\) is 10 units. Given that \(k > 0\), find the value of \(k\).

Given the vectors \(\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ 5 \end{pmatrix}\), calculate the resultant vector of \(2\mathbf{a} + \mathbf{b}\).