Further vectors questions

Questions to attempt

6 questions on Further vectors, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-6.1 · Understand and use the vector and Cartesian forms of an equation of a straight line in 3-D.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

   A line has Cartesian equation $\dfrac{x-1}{2}=\dfrac{y+3}{-1}=\dfrac{z-4}{3}$. Write it in vector form and find the point for which the common parameter is $2$.

   (4)

   (Total for Question 1 is 4 marks)

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2. CP-6.1 · Understand and use the vector and Cartesian forms of an equation of a straight line in 3-D.

   Q2

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   2.

   Write a vector equation of the line through $(2,-1,4)$ with direction vector $3\mathbf i+2\mathbf j-\mathbf k$.

   (2)

   (Total for Question 2 is 2 marks)

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3. CP-6.2 · Understand and use the vector and Cartesian forms of the equation of a plane.

   Q3

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   3.

   The plane $\Pi$ has vector equation $\mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\0\end{pmatrix}+\mu\begin{pmatrix}0\\1\\1\end{pmatrix}$. Find a Cartesian equation of $\Pi$.

   (5)

   (Total for Question 3 is 5 marks)

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4. CP-6.3 · Calculate the scalar product and use it to express the equation of a plane, and to calculate the angle between two lines, two planes and between a line and a plane.

   Q4

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   4.

   Planes $\Pi_1$ and $\Pi_2$ have normals $(1,2,2)$ and $(2,-1,2)$ respectively. Find the exact cosine of the acute angle between the planes.

   (4)

   (Total for Question 4 is 4 marks)

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5. CP-6.4 · Check whether vectors are perpendicular by using the scalar product.

   Q5

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   5.

   Find the value of $k$ for which $(k,2,-1)$ is perpendicular to $(3,1,5)$.

   (3)

   (Total for Question 5 is 3 marks)

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6. CP-6.5 · Find the intersection of a line and a plane. Calculate the perpendicular distance between two lines, from a point to a line and from a point to a plane.

   Q6

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   6.

   Find the perpendicular distance from $P(2,-1,4)$ to the plane $2x-y+2z-9=0$.

   (4)

   (Total for Question 6 is 4 marks)

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CP-6 Further vectors revision notes