A LevelFurther Maths: Core PureEdexcelTopic Questions6. Further Vectors6.2 Vector Planes

Vector Planes (Edexcel A Level Further Maths: Core Pure)

Topic Questions

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1
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The line l subscript 1 has equation fraction numerator x minus 2 over denominator 4 end fraction equals fraction numerator y minus 4 over denominator negative 2 end fraction equals fraction numerator z plus 6 over denominator 1 end fraction

The plane capital pi has equation x minus 2 y space plus space z space equals space 6 space

The line l subscript 2 is the reflection of the line l subscript 1in the plane capital pi

Find a vector equation of the line l subscript 2

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2a
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The plane capital pi subscript 1 has vector equation

bold r. left parenthesis 3 bold i space minus space 4 bold j space plus space 2 bold k right parenthesis space equals space 5

(a)
Find the perpendicular distance from the point (6, 2, 12) to the plane capital pi subscript 1

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2b
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The plane capital pi subscript 2 has vector equation

bold r bold space equals space lambda left parenthesis 2 bold i space plus space bold j space plus space 5 bold k right parenthesis space plus space mu left parenthesis bold i space minus bold space bold j bold space minus space 2 bold k right parenthesis

where lambda and mu are scalar parameters.

(b)
Show that the vector negative bold i space minus space 3 bold j space plus space bold k is perpendicular to capital pi subscript 2

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2c
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(c)
Show that the acute angle between capital pi subscript 1 and capital pi subscript 2is 52° to the nearest degree.

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3a
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The plane capital pi subscript 1 has equation

bold r bold space equals space 2 bold i space plus space 4 bold j space – space bold k space plus space lambda space left parenthesis bold i space plus space 2 bold j space – space 3 bold k right parenthesis space plus space mu left parenthesis – bold i space plus space 2 bold j space plus space bold k right parenthesis

where lambda and mu are scalar parameters.

(a)
Find a Cartesian equation for capital pi subscript 1

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3b
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The line l has equation

fraction numerator x minus 1 over denominator 5 end fraction equals fraction numerator y minus 3 over denominator negative 3 end fraction equals fraction numerator z plus 2 over denominator 4 end fraction

(b)
Find the coordinates of the point of intersection l with capital pi subscript 1

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3c
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The plane capital pi subscript 2 has equation

bold r. left parenthesis 2 bold i space – space bold j space plus space 3 bold k right parenthesis space equals space 5

(c)
Find, to the nearest degree, the acute angle between capital pi subscript 1 and capital pi subscript 2

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4a
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bold italic M equals stretchy left parenthesis table row k 5 7 row 1 1 1 row 2 1 cell negative 1 end cell end table stretchy right parenthesis where k is a constant

(a)
Given that k not equal to 4 comma find, in terms of k, the inverse of the matrix M.

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4b
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(b)
Find, in terms ofspace p, the coordinates of the point where the following planes intersect.

2 x space plus space 5 y space plus space 7 z space equals space 1

x space plus space y space plus space z space equals space p

2 x space plus space y space minus space z space equals space 2

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4c
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(c)
(i)
Find the value of q for which the following planes intersect in a straight line.


4 x space plus space 5 y space plus space 7 z space equals space 1

x space plus space y space plus space z space equals space q

2 x space plus space y space minus space z space equals space 2

(ii)
For this value of q, determine a vector equation for the line of intersection.

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5a
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The line l subscript 1 has equation

fraction numerator x minus 1 over denominator 2 end fraction equals fraction numerator y plus 1 over denominator negative 1 end fraction equals fraction numerator z minus 4 over denominator 3 end fraction

The line l subscript 2 has equation

bold r bold space equals space bold i space plus space 3 bold k space plus space t left parenthesis bold i space minus space bold j space plus space 2 bold k right parenthesis

where t is a scalar parameter.

(a)
Show that l subscript 1 and l subscript 2 lie in the same plane.

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5b
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(b)
Write down a vector equation for the plane containing l subscript 1 and l subscript 2

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6a
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bold M equals open parentheses table row 2 cell negative 1 end cell 1 row 3 k 4 row 3 2 cell negative 1 end cell end table close parentheses where k is a constant

(a)
Find the values of k for which the matrix bold M has an inverse.

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6b
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(b)
Find, in terms of space p, the coordinates of the point where the following planes intersect

2 x space – space y space plus space z space equals space p

3 x space – space 6 y space plus space 4 z space equals space 1

3 x space plus space 2 y space – space z space equals space 0

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6c
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(c)
(i)
Find the value of q for which the set of simultaneous equations

2 x space – space y space plus space z space equals space 1

3 x space – space 5 y space plus space 4 z space equals space q

3 x space plus space 2 y space – space z space equals space 0

can be solved.

(ii)
For this value of q, interpret the solution of the set of simultaneous equations geometrically.

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