Collinearity and ratio division

4 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l }

7 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 0 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ 5 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right)$$

1. In the case where \(A B C\) is a straight line, find the values of \(p\) and \(q\).
2. In the case where angle \(B A C\) is \(90 ^ { \circ }\), express \(q\) in terms of \(p\).
3. In the case where \(p = 3\) and the lengths of \(A B\) and \(A C\) are equal, find the possible values of \(q\).

1. The point \(A\), with coordinates \(( 0 , a , b )\) lies on the line \(l _ { 1 }\), which has equation

$$\mathbf { r } = 6 \mathbf { i } + 19 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } )$$

1. Find the values of \(a\) and \(b\). The point \(P\) lies on \(l _ { 1 }\) and is such that \(O P\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin.
2. Find the position vector of point \(P\). Given that \(B\) has coordinates \(( 5,15,1 )\),
3. show that the points \(A , P\) and \(B\) are collinear and find the ratio \(A P : P B\).

9 Points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) relative to an origin \(O\) in 3-dimensional space. Rectangles \(O A D C\) and \(B E F G\) are the base and top surface of a cuboid. \includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-07_522_812_952_280}

• The point \(M\) is the midpoint of \(B C\).
• The point \(X\) lies on \(A M\) such that \(A X = 2 X M\).
  1. Find \(\overrightarrow { O X }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), simplifying your answer.
  2. Hence show that the lines \(O F\) and \(A M\) intersect.

13. Relative to a fixed origin \(O\)

• the point \(P\) has position vector \(( 0 , - 1,2 )\)
• the point \(Q\) has position vector \(( 1,1,5 )\)
• the point \(R\) has position vector ( \(3,5 , m\) )
  where \(m\) is a constant.
  Given that \(P , Q\) and \(R\) lie on a straight line,
  a. find the value of \(m\)

The line segment \(O Q\) is extended to a point \(T\) so that \(\overrightarrow { R T }\) is parallel to \(\overrightarrow { O P }\) b. Show that \(| \overrightarrow { O T } | = 9 \sqrt { 3 }\).

1. Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(4 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(4 \mathbf { j } + 6 \mathbf { k }\)
• the point \(C\) has position vector \(- 16 \mathbf { i } + p \mathbf { j } + 10 \mathbf { k }\) where \(p\) is a constant.
  Given that \(A , B\) and \(C\) lie on a straight line,
  1. find the value of \(p\).

The line segment \(O B\) is extended to a point \(D\) so that \(\overrightarrow { C D }\) is parallel to \(\overrightarrow { O A }\) (b) Find \(| \overrightarrow { O D } |\), writing your answer as a fully simplified surd.

3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.

5. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.

1. Show that \(A , B\) and \(C\) all lie on a single straight line.
2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
3. Show that \(A D\) is perpendicular to \(B D\).
4. Find the exact area of triangle \(A B D\).
   5. continued

5 Points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\). Point \(C\) lies on \(A B\) such that \(A C : C B = p : 1\).

1. Show that the position vector of \(C\) is \(\frac { 1 } { p + 1 } ( \mathbf { a } + p \mathbf { b } )\). It is now given that \(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) and \(\mathbf { b } = - 6 \mathbf { i } + 4 \mathbf { j } + 12 \mathbf { k }\), and that \(C\) lies on the \(y\)-axis.
2. Find the value of \(p\).
3. Write down the position vector of \(C\).