Advertisements
Advertisements
प्रश्न
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `-5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3:2
(i) internally
(ii) externally
उत्तर
Let `bar("p"), bar("q")` and `bar("r")` be the position vectors of points P, Q and R respectively.
∴ `bar("p") = 2hat"i" - hat"j" + 3hat"k", bar("q") = -5hat"i" + 2hat"j" - 5hat"k"`, m:n = 3:2
(i) R divides the line PQ internally in the ratio 3:2
∴ By using section formula for internal division,
`bar("r") = ("m"bar("q") + "n"bar("p"))/("m" + "n")`
= `(3(-5hat"i" + 2hat"j" - 5hat"k") + 2(2hat"i" - hat"j" + 3hat"k"))/(3 + 2)`
= `(-15hat"i" + 6hat"j" - 15hat"k" + 4hat"i" - 2hat"j" + 6hat"k")/5`
∴ `bar("r") = (-11hat"i" + 4hat"j" - 9hat"k")/5`
= `(-11)/5hat"i" + 4/5hat"j" - 9/5hat"k"`
(ii) R divides the line PQ externally in ratio 3:2
∴ By using section formula for external division,
`bar("r") = ("m"bar("q") - "n"bar("p"))/("m" - "n")`
= `(3(-5hat"i" + 2hat"j" - 5hat"k") - 2(2hat"i" - hat"j" + 3hat"k"))/(3 - 2)`
= `(-15hat"i" + 6hat"j" - 15hat"k" - 4hat"i" + 2hat"j" - 6hat"k")/1`
∴ `bar("r") = -19hat"i" + 8hat"j" - 21hat"k"`
संबंधित प्रश्न
If `bar p = hat i - 2 hat j + hat k and bar q = hat i + 4 hat j - 2 hat k` are position vector (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2:1
If point C `(barc)` divides the segment joining the points A(`bara`) and B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`
Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `P(2veca + vecb)` and `Q(veca - 3vecb)` externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.
If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r), then find the values of p, q, r.
Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
(Pythagoras's Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.
Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.
If AD is the median of ∆ABC, using vectors, prove that \[{AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)\]
In a quadrilateral ABCD, prove that \[{AB}^2 + {BC}^2 + {CD}^2 + {DA}^2 = {AC}^2 + {BD}^2 + 4 {PQ}^2\] where P and Q are middle points of diagonals AC and BD.
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hati - hatj + 3hatk` and `- 5hati + 2hatj - 5hatk` in the ratio 3:2 is internally.
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `- 5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3 : 2 is externally.
Find the position vector of midpoint M joining the points L(7, –6, 12) and N(5, 4, –2).
If the points A(3, 0, p), B(–1, q, 3) and C(–3, 3, 0) are collinear, then find
- the ratio in which the point C divides the line segment AB
- the values of p and q.
The position vector of points A and B are `6bar"a" + 2bar"b"` and `bar"a" - 3bar"b"`. If the point C divides AB in the ratio 3 : 2, show that the position vector of C is `3bar"a" - bar"b"`.
Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.
In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.
Prove that `(bar"a" xx bar"b").(bar"c" xx bar"d")` =
`|bar"a".bar"c" bar"b".bar"c"|`
`|bar"a".bar"d" bar"b".bar"d"|.`
If `bara, barb` and `barc` are position vectors of the points A, B, C respectively and `5bara - 3barb - 2barc = bar0`, then find the ratio in which the point C divides the line segement BA.
If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r
Prove that medians of a triangle are concurrent
Prove that altitudes of a triangle are concurrent
Prove that the angle bisectors of a triangle are concurrent
If A(1, 3, 2), B(a, b, - 4) and C(5, 1, c) are the vertices of triangle ABC and G(3, b, c) is its centroid, then
If the plane 2x + 3y + 5z = 1 intersects the co-ordinate axes at the points A, B, C, then the centroid of Δ ABC is ______.
In a quadrilateral ABCD, M and N are the mid-points of the sides AB and CD respectively. If AD + BC = tMN, then t = ____________.
If G(3, -5, r) is centroid of triangle ABC where A(7, - 8, 1), B(p, q, 5) and C(q + 1, 5p, 0) are vertices of a triangle then values of p, q, rare respectively.
P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then `overline"OA" + overline"OB" + overline"OC" + overline"OD"` = ______
If P(2, 2), Q(- 2, 4) and R(3, 4) are the vertices of Δ PQR then the equation of the median through vertex R is ______.
If the position vectors of points A and B are `hati + 8hatj + 4hatk` and `7hati + 2hatj - 8hatk`, then what will be the position vector of the midpoint of AB?
If G and G' are the centroids of the triangles ABC and A'B'C', then `overline("A""A"^') + overline("B""B"^') + overline("C""C"^')` is equal to ______
If the orthocentre and circumcentre of a triangle are (-3, 5, 1) and (6, 2, -2) respectively, then its centroid is ______
If M and N are the midpoints of the sides BC and CD respectively of a parallelogram ABCD, then `overline(AM) + overline(AN)` = ______
If G`(overlineg)` is the centroid, `H(overlineh)` is the orthocentre and P`(overlinep)` is the circumcentre of a triangle and `xoverlinep + yoverlineh + zoverlineg = 0`, then ______
If A, B, C are the vertices of a triangle whose position vectors are `overline("a"),overline("b"),overline("c")` and G is the centroid of the `triangle ABC,` then `overline("GA")+overline("GB")+overline("GC")` is ______.
The co-ordinates of the points which divides line segment joining the point A(2, –6, 8) and B(–1, 3,–4) internally in the ratio 1: 3' are ______.
Let `square`PQRS be a quadrilateral. If M and N are midpoints of the sides PQ and RS respectively then `bar"PS" + bar"OR"` = ______.
In ΔABC, P is the midpoint of BC, Q divides CA internally in the ratio 2:1 and R divides AB externally in the ratio 1:2, then ______.
Find the unit vector in the diret:tion of the vector `veca = hati + hatj + 2hatk`
What is the midpoint of the vector joining the point P(2, 3, 4) and Q(4, 1, –2)?
In ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)` is equal to ______.
If `overlinea, overlineb, overlinec` are the position vectors of the points A, B, C respectively and `5overlinea + 3overlineb - 8overlinec = overline0` then find the ratio in which the point C divides the line segment AB.
The position vector of points A and B are `6 bar "a" + 2 bar "b" and bar "a" - 3 bar"b"`. If the point C divided AB in the ratio 3 : 2, show that the position vector of C is `3 bar "a" - bar "b".`
If `bara, barb` and `barr` are position vectors of the points A, B and R respectively and R divides the line segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.
Using vector method, prove that the perpendicular bisectors of sides of a triangle are concurrent.
Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be the third point on the line AB dividing the segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.
If `bara, barb, barc` are the position vectors of the points A, B, C respectively and `5 bar a - 3 bar b - 2 bar c = bar 0`, then find the ratio in which the point C divides the line segment BA.
The position vector of points A and B are `6bara + 2 barb and bara - 3 barb`. If point C divides AB in the ratio 3 : 2, then show that the position vector of C is `3bara - barb`.
The position vector of points A and B are `6bara + 2 barb and bara - 3 barb`. If point C divides AB in the ratio 3 : 2, then show that the position vector of C is `3bara - barb`.
The position vector of points A and B are `6bara + 2 barb` and `bara-3 barb`. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is `3bara -barb`.
The position vector of points A and B are `6bara + 2barb` and `bara - 3barb`. If the point C divides AB in the ratio 3 : 2, then show that the position vector of C is `3bara - barb`.
The position vector of points A and B are `6 bara + 2 barb and bara - 3 barb`. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is `3 bara - barb`.
The position vector of points A and B are 6`bara + 2barb and bara - 3barb`. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is 3`bara - barb`.
