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(i) Find a vector equation of the line through the points
A and
B with position vectors
7i+8j+9k and
-i-8j+k respectively.
(ii) The perpendicular to this line from the point
C with position vector
i+8j+3k meets the line at point
N. Find the position vector of
N and the ratio
AN:NB.
iii) Find a Cartesian equation of the line which is a reflection of the line
AC in the line
AB.

(i) Find a vector equation of the line through the points $A$ and $B$ with position vectors $7 \mathbf{i}+8 \mathbf{j}+9 \mathbf{k}$ and $-\mathbf{i}-8 \mathbf{j}+\mathbf{k}$ respectively. $\newline$ (ii) The perpendicular to this line from the point $C$ with position vector $\mathbf{i}+8 \mathbf{j}+3 \mathbf{k}$ meets the line at point $N$ . Find the position vector of $N$ and the ratio $A N: N B$ . $\newline$ iii) Find a Cartesian equation of the line which is a reflection of the line $A C$ in the line $B$ $0$ .

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Find equations of tangent lines using limits

Full solution

Q. (i) Find a vector equation of the line through the points

A

and

B

with position vectors

7 \mathbf{i}+8 \mathbf{j}+9 \mathbf{k}

and

-\mathbf{i}-8 \mathbf{j}+\mathbf{k}

respectively.

\newline

(ii) The perpendicular to this line from the point

C

with position vector

\mathbf{i}+8 \mathbf{j}+3 \mathbf{k}

meets the line at point

N

. Find the position vector of

N

and the ratio

A N: N B

.

\newline

iii) Find a Cartesian equation of the line which is a reflection of the line

A C

in the line

B

0

.

Calculate Direction Vector: To find the vector equation of the line through points $A$ and $B$ , we need the direction vector, which is $B - A$ .\ Direction vector $= (-i - 8j + k) - (7i + 8j + 9k) = -8i - 16j - 8k$ .
Write Vector Equation: The vector equation of a line can be written as $r = a + tb$ , where $a$ is a point on the line, $b$ is the direction vector, and $t$ is a scalar. $\newline$ So, the vector equation of the line through $A$ and $B$ is $r = (7i + 8j + 9k) + t(-8i - 16j - 8k)$ .
Find Position Vector of N: To find the position vector of N, we need to find the value of $t$ when the line through $A$ and $B$ is perpendicular to vector $CN$ . Let's find vector $CN$ first: $CN = N - C$ . Since $N$ lies on the line, $N = A + tb$ . So, $CN = (A + tb) - C$ .
Substitute Given Vectors: Substitute the given vectors into the equation: $CN = ((7i + 8j + 9k) + t(-8i - 16j - 8k)) - (i + 8j + 3k)$ .
Simplify the Equation: Simplify the equation: $CN = (7i + 8j + 9k) + t(-8i - 16j - 8k) - i - 8j - 3k$ . $CN = (6i + t(-8i)) + (16j + t(-16j)) + (6k + t(-8k))$ .
Calculate Dot Product: For CN to be perpendicular to the direction vector, their dot product must be zero. $\newline$ $(6i + t(-8i)) + (16j + t(-16j)) + (6k + t(-8k)) \cdot (-8i - 16j - 8k) = 0$ .
Calculate Dot Product: For CN to be perpendicular to the direction vector, their dot product must be zero. $\newline$ $(6i + t(-8i)) + (16j + t(-16j)) + (6k + t(-8k)) \cdot (-8i - 16j - 8k) = 0$ .Calculate the dot product and set it to zero: $\newline$ $(-48 + 64t) + (-256 + 256t) + (-48 + 64t) = 0$ .

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