Photo AI
Vector a = \begin{pmatrix} 3 \\ -1 \end{pmatrix} and vector b = \begin{pmatrix} -1 \\ 3 \end{pmatrix} - OCR - GCSE Maths - Question 10 - 2019 - Paper 1
Question 10
Vector a = \begin{pmatrix} 3 \\ -1 \end{pmatrix} and vector b = \begin{pmatrix} -1 \\ 3 \end{pmatrix}. (a) Find the values of k and n so that k(a + b) = \begin{pma... show full transcript
Worked Solution & Example Answer:Vector a = \begin{pmatrix} 3 \\ -1 \end{pmatrix} and vector b = \begin{pmatrix} -1 \\ 3 \end{pmatrix} - OCR - GCSE Maths - Question 10 - 2019 - Paper 1
Step 1
Find the values of k and n so that k(a + b) = \begin{pmatrix} 10 \\ n \end{pmatrix}
Answer
To find k and n, we first calculate the vector a + b:
[ a + b = \begin{pmatrix} 3 \ -1 \end{pmatrix} + \begin{pmatrix} -1 \ 3 \end{pmatrix} = \begin{pmatrix} 3 - 1 \ -1 + 3 \end{pmatrix} = \begin{pmatrix} 2 \ 2 \end{pmatrix} ]
Next, we evaluate the expression k(a + b):
[ k(a + b) = k \begin{pmatrix} 2 \ 2 \end{pmatrix} = \begin{pmatrix} 2k \ 2k \end{pmatrix} ]
Setting this equal to \begin{pmatrix} 10 \ n \end{pmatrix}, we have two equations:
-
[ 2k = 10 ] Which gives us [ k = 5 ]
-
[ 2k = n ] Substituting the value of k, we find [ n = 2(5) = 10 ]
Thus, the values are:
- k = 5
- n = 10
Step 2
Gavin starts to draw a diagram to show that a + 2b = \begin{pmatrix} 5 \\ 5 \end{pmatrix}
Answer
First, let's compute 2b:
[ 2b = 2 \begin{pmatrix} -1 \ 3 \end{pmatrix} = \begin{pmatrix} -2 \ 6 \end{pmatrix} ]
Now, we add vector a and 2b:
[ a + 2b = \begin{pmatrix} 3 \ -1 \end{pmatrix} + \begin{pmatrix} -2 \ 6 \end{pmatrix} = \begin{pmatrix} 3 - 2 \ -1 + 6 \end{pmatrix} = \begin{pmatrix} 1 \ 5 \end{pmatrix} ]
It appears that the result does not equal \begin{pmatrix} 5 \ 5 \end{pmatrix}, thus, Gavin would need to adjust the diagram to correctly represent the relationship.