Vectors：練習問題

The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are given by \(\vec{OA} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\) and \(\vec{OB} = 4\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). Find the magnitude of the displacement vector \(\vec{AB}\).

Two lines \(L_1\) and \(L_2\) have equations:
\(L_1: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\)
\(L_2: \mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}\)
Show that the lines intersect and find the coordinates of their point of intersection.

The lines \(l_1\) and \(l_2\) have vector equations:
\(l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}\)
\(l_2: \mathbf{r} = \begin{pmatrix} 3 \\ 5 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\)
Determine whether the lines intersect, and if so, find the coordinates of the point of intersection.

Find the coordinates of the midpoint of the line segment joining the points with position vectors \( \mathbf{a} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} \) and \( \mathbf{b} = \mathbf{i} + 4\mathbf{j} - \mathbf{k} \).

Find the angle between the vectors \(\mathbf{a} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{b} = 3\mathbf{i} + 4\mathbf{k}\), giving your answer correct to the nearest 0.1 degree.

Find the unit vector in the direction of $$\mathbf{v} = 6\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$$.

The vector position of point \(A\) is \(2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\) and the vector position of point \(B\) is \(\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). Find the position vector of point \(M\) such that \(M\) is the midpoint of the line segment \(AB\).

Relative to an origin \(O\), the point \(A\) has position vector \(2\mathbf{i} + p\mathbf{j} - \mathbf{k}\) and point \(B\) has position vector \(4\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}\). If the magnitude of the vector \(\vec{AB}\) is \(7\) units, find the possible values of the constant \(p\).

The line \( l_1 \) passes through the point \( A(1, 2, -1) \) and is parallel to the vector \( \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} \). The line \( l_2 \) passes through the point \( B(3, 0, 4) \) and is parallel to the vector \( \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix} \).
(a) Determine the coordinates of the point of intersection of \( l_1 \) and \( l_2 \).
(b) Calculate the acute angle between the directions of \( l_1 \) and \( l_2 \), giving your answer in degrees correct to 1 decimal place.

A tetrahedron has vertices \(O(0, 0, 0)\), \(A(6, 0, 0)\), \(B(0, 4, 0)\), and \(C(0, 0, 12)\). The point \(M\) is the midpoint of the edge \(AC\).
(a) Find the position vector of \(M\).
(b) Use the scalar product to find the angle between the line \(BM\) and the edge \(BC\).
(c) Find the vector equation of the line passing through \(O\) that is parallel to the vector \(\vec{BC}\).
(d) Determine if the point \(P(0, -2, 6)\) lies on the line found in part (c).