Find the position vectors of a point R which divides the line joining two points P and Q whose position vectors are 2a-3b- and a-3b- respectively- externally in the ratio 1-2- Also- show that P is the midpoint of the line segment R-

R.E.F image #160; R⃗ = 2×P⃗ 1×Q⃗2 1 = 4a⃗+6b⃗ a⃗+3b⃗ = 3a⃗+9b⃗ Mid point of QR⃗ = Q⃗+R⃗2 = (a⃗ 3b⃗+(3a⃗+9b⃗))2 = 2a⃗+3b⃗ = P⃗ ...

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Standard XII

Mathematics

Distance Formula

Find the posi...

Question

Find the position vectors of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $2 \to a + 3 \to b$ and $\to a - 3 \to b$ respectively, externally in the ratio $1 : 2$ . Also, show that $P$ is the midpoint of the line segment $R$ .

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Solution

R.E.F image
$\to R = \frac{2 \times \to P - 1 \times \to Q}{2 - 1}$
$= 4 \to a + 6 \to b - \to a + 3 \to b = 3 \to a + 9 \to b$
Mid-point of $Q \to R = \frac{\to Q + \to R}{2} = \frac{(\to a - 3 \to b + (3 \to a + 9 \to b))}{2}$
$= 2 \to a + 3 \to b = \to P$

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Find the position vectors of a point R which divides the line joining two points P and Q whose position vectors are 2→a+3→b and →a−3→b respectively, externally in the ratio 1:2. Also, show that P is the midpoint of the line segment R.

R.E.F image →R=2×→P−1×→Q2−1=4→a+6→b−→a+3→b=3→a+9→bMid-point of Q→R=→Q+→R2=(→a−3→b+(3→a+9→b))2=2→a+3→b=→P

Find the position vectors of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $2 \to a + 3 \to b$ and $\to a - 3 \to b$ respectively, externally in the ratio $1 : 2$ . Also, show that $P$ is the midpoint of the line segment $R$ .

R.E.F image
$\to R = \frac{2 \times \to P - 1 \times \to Q}{2 - 1}$
$= 4 \to a + 6 \to b - \to a + 3 \to b = 3 \to a + 9 \to b$
Mid-point of $Q \to R = \frac{\to Q + \to R}{2} = \frac{(\to a - 3 \to b + (3 \to a + 9 \to b))}{2}$
$= 2 \to a + 3 \to b = \to P$