Vector between two points

4 It is given that \(A B C D\) is a quadrilateral. The position vector of \(A\) is \(\mathbf { i } + \mathbf { j }\), and the position vector of \(B\) is \(3 \mathbf { i } + 5 \mathbf { j }\).

1. Find the length \(A B\).
2. The position vector of \(C\) is \(p \mathbf { i } + p \mathbf { j }\) where \(p\) is a constant greater than 1 . Given that the length \(A B\) is equal to the length \(B C\), determine the position vector of \(C\).
3. The point \(M\) is the midpoint of \(A C\). Given that \(\overrightarrow { M D } = 2 \overrightarrow { B M }\), determine the position vector of \(D\).
4. State the name of the quadrilateral \(A B C D\), giving a reason for your answer.

1. Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
   (b) find \(| \overrightarrow { A B } |\).

Give your answer as a simplified surd.

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

• point \(A\) has position vector \(10 \mathbf { i } - 3 \mathbf { j }\)
• point \(B\) has position vector \(- 8 \mathbf { i } + 9 \mathbf { j }\)
• point \(C\) has position vector \(- 2 \mathbf { i } + p \mathbf { j }\) where \(p\) is a constant
  1. Find \(\overrightarrow { A B }\)
  2. Find \(| \overrightarrow { A B } |\) giving your answer as a fully simplified surd.

Given that points \(A , B\) and \(C\) lie on a straight line,

1. find the value of \(p\),
2. state the ratio of the area of triangle \(A O C\) to the area of triangle \(A O B\).

5 Points \(A , B , C\) and \(D\) have position vectors \(\mathbf { a } = \binom { 1 } { 2 } , \mathbf { b } = \binom { 3 } { 5 } , \mathbf { c } = \binom { 7 } { 4 }\) and \(\mathbf { d } = \binom { 4 } { k }\).

1. Find the value of \(k\) for which \(D\) is the midpoint of \(A C\).
2. Find the two values of \(k\) for which \(| \overrightarrow { A D } | = \sqrt { 13 }\).
3. Find one value of \(k\) for which the four points form a trapezium.

3 The points \(A\) and \(B\) have position vectors \(\binom { 2 } { - 1 }\) and \(\binom { 5 } { 4 }\) respectively. The vector \(\overrightarrow { \mathrm { AC } }\) is \(\binom { - 2 } { 2 }\).

1. Write down the position vector of C as a column vector.
2. Show that B is equidistant from A and C .

6 Two points, \(A\) and \(B\), have position vectors \(\mathbf { a } = \mathbf { i } - 3 \mathbf { j }\) and \(\mathbf { b } = 4 \mathbf { i } + 3 \mathbf { j }\).
The point C lies on the line \(y = 1\). The lengths of the line segments AC and BC are equal. Determine the position vector of \(C\).

4 The position vector of \(P\) is \(\mathbf { p } = \binom { 4 } { 3 }\) and the position vector of \(Q\) is \(\mathbf { q } = \binom { 28 } { 10 }\).

1. Determine the magnitude of \(\overrightarrow { \mathrm { PQ } }\).
2. Determine the angle between \(\overrightarrow { \mathrm { PQ } }\) and the positive \(x\)-direction.

8 With respect to an origin O , the position vectors of the points A and B are \(\overrightarrow { \mathrm { OA } } = \binom { - 3 } { 20 }\) and \(\overrightarrow { \mathrm { OB } } = \binom { 6 } { 8 }\).

1. Determine whether \(| \overrightarrow { \mathrm { AB } } | > 200\). The point C is such that \(\overrightarrow { \mathrm { AC } } = \binom { 18 } { - 24 }\).
2. Determine whether \(\mathrm { A } , \mathrm { B }\) and C are collinear.

5 You are given that \(\overrightarrow { \mathrm { OA } } = \binom { 3 } { - 1 }\) and \(\overrightarrow { \mathrm { OB } } = \binom { 5 } { - 3 }\). Determine the exact length of \(A B\).

1 Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by \(\mathbf { u } = \binom { 4 } { 6 }\) and \(\mathbf { v } = \binom { - 3 } { 2 }\).

1. Find \(\mathbf { u } + \mathbf { v }\) and \(\mathbf { u } - \mathbf { v }\).
2. Show that \(| \mathbf { u } + \mathbf { v } | = | \mathbf { u } - \mathbf { v } |\).

One of the following is an expression for the distance between the points represented by position vectors \(5\mathbf{i} - 3\mathbf{j}\) and \(18\mathbf{i} + 7\mathbf{j}\) Identify the correct expression. Tick (\(\checkmark\)) one box. [1 mark] \(\sqrt{13^2 + 4^2}\) \(\sqrt{13^2 + 10^2}\) \(\sqrt{23^2 + 4^2}\) \(\sqrt{23^2 + 10^2}\)

Given that the point \(A\) has position vector \(3\mathbf{i} - 7\mathbf{j}\) and the point \(B\) has position vector \(8\mathbf{i} + 3\mathbf{j}\).

1. find the vector \(\overrightarrow{AB}\) [2]
2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]

The position vectors of the points A and B, relative to a fixed origin O, are given by $$\mathbf{a} = 4\mathbf{i} + 7\mathbf{j}, \quad\quad \mathbf{b} = \mathbf{i} + 3\mathbf{j},$$ respectively.

1. Find the vector \(\overrightarrow{AB}\). [2]
2. Determine the distance between the points A and B. [2]
3. The position vector of the point C is given by \(\mathbf{c} = -2\mathbf{i} + 5\mathbf{j}\). The point D is such that the distance between C and D is equal to the distance between A and B, and \(\overrightarrow{CD}\) is parallel to \(\overrightarrow{AB}\). Find the possible position vectors of the point D. [4]

Given that the point \(A\) has position vector \(3\mathbf{i} - 7\mathbf{j}\) and the point \(B\) has position vector \(8\mathbf{i} + 3\mathbf{j}\),

1. find the vector \(\overrightarrow{AB}\) [2]
2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]

Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(-2\mathbf{i} + 3\mathbf{j}\),
• the point \(B\) has position vector \(3\mathbf{i} + p\mathbf{j}\), where \(p\) is constant,

Given that \(|\overrightarrow{AB}| = 5\sqrt{2}\), find the possible values for \(p\). [3]

The points \(A\), \(B\) and \(C\) have position vectors \(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \end{pmatrix}\) respectively. \(M\) is the midpoint of \(BC\).

1. Find the position vector of the point \(D\) such that \(\overrightarrow{BC} = \overrightarrow{AD}\). [3]
2. Find the magnitude of \(\overrightarrow{AM}\). [3]