CAIE A-Level Mathematics 3.7.6 Scalar Product

Practise calculating scalar products to find 3D angles, impose perpendicular conditions and locate the foot from a point to a line before deriving exact lengths or areas.

[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 $t=\frac{7}{9}$
A1
Carry out correct method to find O P
DM1
Obtain $\frac{\sqrt{5}}{3}$
A1
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
A1
Use Pythagoras to obtain O N
DM1
Obtain the given answer correctly
A1

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3.7.6—Scalar product question 1

Question (a)

Question (b)