CAIE A-Level Mathematics A2 3.7.6 Scalar Product

[Maximum number: 6]

$\mathbf{9} \quad$ Two lines l and m have equations r=3 i+2 j+5 k+s(4 i-j+3 k) and r=i-j-2 k+t(-i+2 j+2 k) respectively.

(a)

Show that l and m are perpendicular.

[ 2 ]

(b)

Show that the length of the perpendicular from the origin to the line m is $\frac{1}{3} \sqrt{5}$ .

[ 4 ]

Taking a general point P on m, form an equation in t by either equating a
relevant scalar product to zero, or equating the derivative of $|\overrightarrow{O P}|$ to zero, or
taking a specific point Q on m, e.g. (1,-1,-2), using Pythagoras in triangle
O P Q
*M1
e.g. $\left(\begin{array}{c}1-t -1+2 t -2+2 t\end{array}\right) \cdot\left(\begin{array}{c}-1 2 2\end{array}\right)=0$
Obtain $t=\frac{7}{9}$
Carry out correct method to find O P
Obtain $\frac{\sqrt{5}}{3}$
Obtain the given answer from full and correct working.
Alternative method for question 9(c)
Take a specific point Q on m, e.g. ( -1,3,2 ) and use a scalar product to find
Q N, the projection of O Q on m
*M1
Obtain $Q N=\frac{11}{3}$ , or equivalent
Use Pythagoras to obtain O N
Obtain the given answer correctly

CAIE A-Level Mathematics A2 3.7.6 Scalar Product Questions

Exam points

CAIE A-Level Mathematics A2 3.7.6 Scalar Product Questions question 1

Question (a)

Question (b)