Linear combinations of vectors - Vectors 3D & Lines

CAIE P1 2007 June Q9

8 marks Moderate -0.8

9 Relative to an origin $O$, the position vectors of the points $A$ and $B$ are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 4 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ 2 \\ - 4 \end{array} \right) .$$

Given that $C$ is the point such that $\overrightarrow { A C } = 2 \overrightarrow { A B }$, find the unit vector in the direction of $\overrightarrow { O C }$. The position vector of the point $D$ is given by $\overrightarrow { O D } = \left( \begin{array} { l } 1 \\ 4 \\ k \end{array} \right)$, where $k$ is a constant, and it is given that $\overrightarrow { O D } = m \overrightarrow { O A } + n \overrightarrow { O B }$, where $m$ and $n$ are constants.
Find the values of $m , n$ and $k$.

OCR MEI C4 Q1

6 marks Moderate -0.5

1 The points $\mathrm { A } , \mathrm { B }$ and C have coordinates $\mathrm { A } ( 3,2 , - 1 ) , \mathrm { B } ( - 1,1,2 )$ and $\mathrm { C } ( 10,5 , - 5 )$, relative to the origin O . Show that $\overrightarrow { \mathrm { OC } }$ can be written in the form $\lambda \overrightarrow { \mathrm { OA } } + \mu \overrightarrow { \mathrm { OB } }$, where $\lambda$ and $\mu$ are to be determined. What can you deduce about the points $\mathrm { O } , \mathrm { A } , \mathrm { B }$ and C from the fact that $\overrightarrow { \mathrm { OC } }$ can be expressed as a combination of $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OB } }$ ?

OCR MEI C4 2009 January Q3

5 marks Moderate -0.8

3 Vectors $\mathbf { a }$ and $\mathbf { b }$ are given by $\mathbf { a } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }$ and $\mathbf { b } = 4 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$.
Find constants $\lambda$ and $\mu$ such that $\lambda \mathbf { a } + \mu \mathbf { b } = 4 \mathbf { j } - 3 \mathbf { k }$.

OCR MEI C4 2013 June Q5

6 marks Moderate -0.3

5 The points $\mathrm { A } , \mathrm { B }$ and C have coordinates $\mathrm { A } ( 3,2 , - 1 ) , \mathrm { B } ( - 1,1,2 )$ and $\mathrm { C } ( 10,5 , - 5 )$, relative to the origin O . Show that $\overrightarrow { \mathrm { OC } }$ can be written in the form $\lambda \overrightarrow { \mathrm { OA } } + \mu \overrightarrow { \mathrm { OB } }$, where $\lambda$ and $\mu$ are to be determined. What can you deduce about the points $\mathrm { O } , \mathrm { A } , \mathrm { B }$ and C from the fact that $\overrightarrow { \mathrm { OC } }$ can be expressed as a combination of $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OB } }$ ?

CAIE FP1 2014 June Q1

5 marks Moderate -0.5

1 The vectors $\mathbf { a } , \mathbf { b } , \mathbf { c }$ and $\mathbf { d }$ in $\mathbb { R } ^ { 3 }$ are given by $$\mathbf { a } = \left( \begin{array} { r }

CAIE FP1 2018 November Q1

5 marks Moderate -0.5

1 The vectors $\mathbf { a } , \mathbf { b } , \mathbf { c }$ and $\mathbf { d }$ in $\mathbb { R } ^ { 3 }$ are given by $$\mathbf { a } = \left( \begin{array} { l } 1

CAIE FP1 2018 November Q2

6 marks Moderate -0.5

2
1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2
9
0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l }

OCR Stats 1 2018 September Q8

9 marks Moderate -0.5

8 The points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$, relative to an origin $O$, in three dimensions. The figure $O A P B S C T U$ is a cuboid, with vertices labelled as in the following diagram. $M$ is the midpoint of $A U$. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_557_1221_2087_420}