Magnitude of a vector：練習問題

Find the magnitude of the vector \(\mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).

In the coordinate plane, point \(A\) has coordinates \((1, 2)\) and point \(B\) has coordinates \((4, 6)\).
Find the magnitude of the vector \(\vec{AB}\).

A square \(ABCD\) is drawn on a coordinate grid where point \(A\) is at \((0, 0)\) and point \(B\) is at \((3, 4)\).
Find the magnitude of the diagonal vector \(\vec{AC}\).

The vector \(\mathbf{a}\) is given by \(\begin{pmatrix} -6 \\ 8 \end{pmatrix}\).
Calculate \(|\mathbf{a}|\).

The magnitude of the vector \(\mathbf{v} = \begin{pmatrix} k \\ 12 \end{pmatrix}\) is \(13\).
Given that \(k > 0\), find the value of \(k\).

A vector is defined as \(\vec{PQ} = \begin{pmatrix} -8 \\ 15 \end{pmatrix}\). Calculate the magnitude of \(\vec{PQ}\), denoted by \(|\vec{PQ}|\).

The vector \(\mathbf{v} = \begin{pmatrix} k \\ 6 \end{pmatrix}\) has a magnitude of \(10\). Given that \(k < 0\), find the value of \(k\) and write down the unit vector in the direction of \(\mathbf{v}\).

The position vectors of points \(A\) and \(B\) are given by \(\mathbf{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}\) respectively.

(a) Find the vector \(\vec{AB}\) as a column vector.

(b) Calculate the magnitude of the vector \(2\mathbf{a} - \mathbf{b}\), giving your answer in the form \(k\sqrt{2}\) where \(k\) is an integer.

(c) Point \(C\) has position vector \(\mathbf{c} = \begin{pmatrix} x \\ 5 \end{pmatrix}\). Given that the magnitude of vector \(\vec{AC}\) is 5, find the two possible values of the constant \(x\).