Advertisements
Advertisements
प्रश्न
Find the position vector of a point R which divides the line joining the two points P and Q with position vectors \[\vec{OP} = 2 \vec{a} + \vec{b}\] and \[\vec{OQ} = \vec{a} - 2 \vec{b}\], respectively in the ratio 1 : 2 internally and externally.
उत्तर
It is given that P and Q are two points with position vectors\[\vec{OP} = 2 \vec{a} + \vec{b}\] and \[\vec{OP} = 2 \vec{a} + \vec{b}\], respectively.
When R divides PQ internally in the ratio 1 : 2, then
Position vector of R = \[\frac{1 \times \left( \vec{a} - 2 \vec{b} \right) + 2 \times \left( 2 \vec{a} + \vec{b} \right)}{1 + 2} = \frac{5 \vec{a}}{3}\]
When R divides PQ externally in the ratio 1 : 2, then Position vector of R =\[\frac{1 \times \left( \vec{a} - 2 \vec{b} \right) - 2 \times \left( 2 \vec{a} + \vec{b} \right)}{1 - 2} = \frac{- 3 \vec{a} - 4 \vec{b}}{- 1} = 3 \vec{a} + 4 \vec{b}\]
APPEARS IN
संबंधित प्रश्न
Let \[\vec{a,} \vec{b,} \vec{c,} \vec{d}\] be the position vectors of the four distinct points A, B, C, D. If \[\vec{b} - \vec{a} = \vec{c} - \vec{d}\], then show that ABCD is a parallelogram.
Show that the four points A, B, C, D with position vectors \[\vec{a,} \vec{b,} \vec{c,} \vec{d}\] respectively such that \[3 \vec{a} - 2 \vec{b} + 5 \vec{c} - 6 \vec{d} = 0,\] are coplanar. Also, find the position vector of the point of intersection of the line segments AC and BD.
Show that the four points P, Q, R, S with position vectors \[\vec{p}\], \[\vec{q}\], \[\vec{r}\], \[\vec{s}\] respectively such that 5 \[\vec{p}\] − 2 \[\vec{q}\] + 6 \[\vec{r}\] − 9 \[\vec{s}\] \[\vec{0}\], are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.
Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.
If the position vector of a point (−4, −3) be \[\vec{a,}\] find \[\left| \vec{a} \right|\]
If the position vector \[\vec{a}\] of a point (12, n) is such that \[\left| \vec{a} \right|\] = 13, find the value (s) of n.
Show that the points 2 \[\hat{i}, - \hat{i}-4 \] \[\hat{j}\] and \[-\hat{i}+4\hat{j}\] form an isosceles triangle.
The position vectors of points A, B and C are \[\lambda \hat{i} +\] 3 \[\hat{j}\],12\[\hat{i} + \mu\] \[\hat{j}\] and 11\[\hat{i} -\] 3 \[\hat{j}\] respectively. If C divides the line segment joining A and B in the ratio 3:1, find the values of \[\lambda\] and \[\mu\]
Find a unit vector in the direction of the resultant of the vectors
\[\hat{i} - \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} - 2 \hat{k} \text{ and }\hat{i} + 2 \hat{j} - 2 \hat{k} .\]
If \[\overrightarrow{PQ} = 3 \hat{i} + 2 \hat{j} - \hat{k}\] and the coordinates of P are (1, −1, 2), find the coordinates of Q.
Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q(4, 1, −2).
Show that the points A, B, C with position vectors \[\vec{a} - 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + 3 \vec{b} - 4 \vec{c}\] and \[- 7 \vec{b} + 10 \vec{c}\] are collinear.
If the points with position vectors \[10 \hat{i} + 3 \hat{j} , 12 \hat{i} - 5 \hat{j}\text{ and a }\hat{i} + 11 \hat{j}\] are collinear, find the value of a.
If the points A(m, −1), B(2, 1) and C(4, 5) are collinear, find the value of m.
Define position vector of a point.
If D is the mid-point of side BC of a triangle ABC such that \[\overrightarrow{AB} + \overrightarrow{AC} = \lambda \overrightarrow{AD} ,\] write the value of λ.
Find the image P' of the point P having position vector `hati+ 3hatj+ 4hatk` in the plane `vecr. (2hati - hatj + hatk) + 3 = 0 .` Hence find the length of PP'.
Find the position vector of the point which divides the join of points with position vectors `vec"a" + 3vec"b" and vec"a"- vec"b"` internally in the ratio 1 : 3.
X and Y are two points with position vectors `3vec("a") + vec("b")` and `vec("a")-3vec("b")`respectively. Write the position vector of a point Z which divides the line segment XY in the ratio 2 : 1 externally.
Find the value of x such that the four-point with position vectors,
`"A"(3hat"i"+2hat"j"+hat"k"),"B" (4hat"i"+"x"hat"j"+5hat"k"),"c" (4hat"i"+2hat"j"-2hat"k")`and`"D"(6hat"i"+5hat"j"-hat"k")`are coplaner.
Find the position vector of a point R which divides the line joining the two points P and Q with position vectors `vec"OP" = 2vec"a" + vec"b"` and `vec"OQ" = vec"a" - 2vec"b"`, respectively, in the ratio 1:2 internally
The position vector of the point which divides the join of points with position vectors `vec"a" + vec"b"` and 2`vec"a" - vec"b"` in the ratio 1:2 is ______.
The position vector of the point which divides the join of points `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1 is ______.
