Find-the-image-of-point-OP-p-in-the-line-r-a-b-

Question Number 104035 by ajfour last updated on 19/Jul/20

F i n d t h e i m a g e o f p o i n t \vec{O P} = \bar{p} i n

t h e l i n e \bar{r} = \bar{a} + λ \bar{b} .

Answered by mr W last updated on 19/Jul/20

Commented by mr W last updated on 19/Jul/20

A N = λ b

\frac{(p - a) \cdot b}{∣ b ∣} = λ ∣ b ∣

λ = \frac{(p - a) \cdot b}{∣ b ∣^{2}}

P N = a + λ b - p

O Q = q = p + 2 P N = p + 2 (a + λ b - p)

= 2 (a + λ b) - p

\Rightarrow q = 2 (a + \frac{(p - a) \cdot b}{∣ b ∣^{2}} b) - p

Commented by ajfour last updated on 19/Jul/20

I b e l i e v e {t h i s}^{'} l l b e t r u e f o r 3 D e v e n

S i r ?

Commented by mr W last updated on 19/Jul/20

b a s i c a l l y y e s .

Commented by ajfour last updated on 19/Jul/20

\frac{p + q}{2} = a + λ b \Rightarrow q = 2 (a + λ b) - p

N o w (q - p) . b = 0

[(a + λ b) - p] . b = 0

λ = \frac{(a - p) . b}{∣ b ∣^{2}}

\bar{q} = (2 \bar{a} - \bar{p}) + (\frac{2 (\bar{a} - \bar{p}) . \bar{b}}{∣ \bar{b} ∣^{2}}) \bar{b}

Commented by mr W last updated on 19/Jul/20

y e s, t h a n k s!

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