Foot of perpendicular from origin to line

1. (i) Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 5 \\ 1 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$ The point \(P\) lies on \(l _ { 1 }\). Given that \(\overrightarrow { O P }\) is perpendicular to \(l _ { 1 }\), calculate the coordinates of \(P\).
(ii) Relative to a fixed origin \(O\), the line \(l _ { 2 }\) is given by the equation $$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ - 3 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 3 \\ 4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).

6. The line \(l _ { 1 }\) has vector equation $$\mathbf { r } = 8 \mathbf { i } + 12 \mathbf { j } + 14 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) ,$$ where \(\lambda\) is a parameter. The point \(A\) has coordinates (4, 8, a), where \(a\) is a constant. The point \(B\) has coordinates ( \(b , 13,13\) ), where \(b\) is a constant. Points \(A\) and \(B\) lie on the line \(l _ { 1 }\).

1. Find the values of \(a\) and \(b\). Given that the point \(O\) is the origin, and that the point \(P\) lies on \(l _ { 1 }\) such that \(O P\) is perpendicular to \(l _ { 1 }\),
2. find the coordinates of \(P\).
3. Hence find the distance \(O P\), giving your answer as a simplified surd.

4 Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.

1. Find a vector equation of the line passing through \(A\) and \(B\).
2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).

8. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.

1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
3. Find the coordinates of \(D\).
4. Find the area of triangle \(O A B\) to 3 significant figures.

9 The equation of a straight line \(l\) is \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) . O\) is the origin.

1. The point \(P\) on \(l\) is given by \(t = 1\). Calculate the acute angle between \(O P\) and \(l\).
2. Find the position vector of the point \(Q\) on \(l\) such that \(O Q\) is perpendicular to \(l\).
3. Find the length of \(O Q\).

5 The line through points \(A ( 8 , - 7 , - 2 )\) and \(B ( 11 , - 9,0 )\) is denoted by \(L _ { 1 }\).

1. Find a vector equation for \(L _ { 1 }\).
2. Determine whether the point \(( 26 , - 19 , - 14 )\) lies on \(L _ { 1 }\). The line \(L _ { 2 }\) passes through the origin, \(O\), and intersects \(L _ { 1 }\) at the point \(C\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) are perpendicular.
3. By using the fact that \(C\) lies on \(L _ { 1 }\), find a vector equation for \(L _ { 2 }\).
4. Hence find the shortest distance from \(O\) to \(L _ { 1 }\).

4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }\), and the point \(B\) has position vector \(7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }\).

1. Find the vector \(\overrightarrow { A B }\).
2. Calculate the cosine of \(\angle O A B\).
3. Show that, for all values of \(\lambda\), the point P with position vector \(\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }\) lies on the line through \(A\) and \(B\).
4. Find the value of \(\lambda\) for which \(O P\) is perpendicular to \(A B\).
5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(A B\).

3. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = ( \mathbf { i } + p \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 3 \mathbf { i } - \mathbf { j } + q \mathbf { k } ) ,$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter.
Given that the point \(A\) with coordinates \(( - 5,9 , - 9 )\) lies on \(l\),

1. find the values of \(p\) and \(q\),
2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
3. Find the coordinates of \(C\).
4. Find the ratio \(A C : C B\) 3. continued

4. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.

1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
3. Find the coordinates of \(D\).
4. Find the area of triangle \(O A B\) to 3 significant figures.
   4. continued

With respect to a fixed origin \(O\), the line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$ where \(\mu\) is a scalar parameter. The point \(A\) is on \(l_1\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). [1] The acute angle between \(OA\) and \(l_1\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). [3] The point \(B\) is such that \(\overrightarrow{OB} = 3\overrightarrow{OA}\). The line \(l_2\) passes through the point \(B\) and is parallel to the line \(l_1\).
3. Find a vector equation of \(l_2\). [2]
4. Find the length of \(OB\), giving your answer as a simplified surd. [1] The point \(X\) lies on \(l_2\). Given that the vector \(\overrightarrow{OX}\) is perpendicular to \(l_2\),
5. find the length of \(OX\), giving your answer to 3 significant figures. [3]

Relative to a fixed origin \(O\), the point \(A\) has position vector \(4\mathbf{i} + 8\mathbf{j} - \mathbf{k}\), and the point \(B\) has position vector \(7\mathbf{i} + 14\mathbf{j} + 5\mathbf{k}\).

1. Find the vector \(\overrightarrow{AB}\). [1]
2. Calculate the cosine of \(\angle OAB\). [3]
3. Show that, for all values of \(\lambda\), the point \(P\) with position vector $$\lambda\mathbf{i} + 2\lambda\mathbf{j} + (2\lambda - 9)\mathbf{k}$$ lies on the line through \(A\) and \(B\). [3]
4. Find the value of \(\lambda\) for which \(OP\) is perpendicular to \(AB\). [3]
5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(AB\). [2]

Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((9\mathbf{i} - 2\mathbf{j} + \mathbf{k})\), \((6\mathbf{i} + 2\mathbf{j} + 6\mathbf{k})\) and \((3\mathbf{i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.

1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]

Given that \(C\) lies on \(l\),

1. find the value of \(p\) and the value of \(q\), [2]
2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]

The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).

1. Find the position vector of \(D\). [6]

Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((\mathbf{9i} - \mathbf{2j} + \mathbf{k})\), \((\mathbf{6i} + \mathbf{2j} + \mathbf{6k})\) and \((\mathbf{3i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.

1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]

Given that \(C\) lies on \(l\),

1. find the value of \(p\) and the value of \(q\), [2]
2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]

The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).

1. Find the position vector of \(D\). [6]

The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5\mathbf{i} + 2\mathbf{j} - 9\mathbf{k}\) and \(4\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) respectively.

1. Find a vector equation for the line \(PQ\). [2]

The position vector of the point \(T\) is \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).

1. Write down a vector equation for the line \(OT\) and show that \(OT\) is perpendicular to \(PQ\). [4]

It is given that \(OT\) intersects \(PQ\).

1. Find the position vector of the point of intersection of \(OT\) and \(PQ\). [3]
2. Hence find the perpendicular distance from \(O\) to \(PQ\), giving your answer in an exact form. [2]

Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf{r} = \begin{pmatrix} 1 \\ p \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -1 \\ q \end{pmatrix},$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter. Given that the point \(A\) with coordinates \((-5, 9, -9)\) lies on \(l\),

1. find the values of \(p\) and \(q\), [3]
2. show that the point \(B\) with coordinates \((25, -1, 11)\) also lies on \(l\). [2]

The point \(C\) lies on \(l\) and is such that \(OC\) is perpendicular to \(l\).

1. Find the coordinates of \(C\). [3]
2. Find the ratio \(AC : CB\) [2]

The points \(O\) and \(A\) have position vectors \(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 0 \\ 8 \end{pmatrix}\) respectively. The point \(P\) is such that \(\overrightarrow{OP} = k\overrightarrow{OA}\), where \(k\) is a non-zero constant.

1. Find, in terms of \(k\), the length of \(OP\). [1] Point \(B\) has position vector \(\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) and angle \(OPB\) is a right angle.
2. Determine the value of \(k\). [4]

The position vector of point \(A\) is \(\mathbf{a} = -9\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}\). The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf{a}\).

1. Determine the shortest distance between the origin, \(O\), and \(l\). [2] \(l\) is also perpendicular to the vector \(\mathbf{b}\) where \(\mathbf{b} = -2\mathbf{i} + \mathbf{j} + \mathbf{k}\).
2. Find a vector which is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). [1]
3. Write down an equation of \(l\) in vector form. [1] \(P\) is a point on \(l\) such that \(PA = 2OA\).
4. Find angle \(POA\) giving your answer to 3 significant figures. [3] \(C\) is a point whose position vector, \(\mathbf{c}\), is given by \(\mathbf{c} = p\mathbf{a}\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf{r} = \mathbf{c} + \mu\mathbf{b}\). The point with position vector \(9\mathbf{i} + 8\mathbf{j} - 12\mathbf{k}\) lies on \(m\).
5. Find the value of \(p\). [3]

The points \(A\) and \(B\) have position vectors \(5\mathbf{j} + 11\mathbf{k}\) and \(c\mathbf{i} + d\mathbf{j} + 21\mathbf{k}\) respectively, where \(c\) and \(d\) are constants. The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf{r} = 5\mathbf{j} + 11\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + 5\mathbf{k})\), where \(\lambda\) is a parameter.

1. Find the value of \(c\) and the value of \(d\). [3]

The point \(P\) lies on the line \(l\), and \(\overrightarrow{OP}\) is perpendicular to \(l\), where \(O\) is the origin.

1. Find the position vector of \(P\). [6]
2. Find the area of triangle \(OAB\), giving your answer to 3 significant figures. [4]