Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let `bara, barb,barc` be the position vectors of `triangle ABC` whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r)
`:.bara = 2hati + pbarj - 3bark`, `barb = qbari - 2barj + 5bark`, `barc = -5bari + barj + rbark`
Given that origin O is the centroid of `triangle ABC`
`:. barO = (bara+barb+barc)/3`
`:. bara+barb+barc = barO`
`2hati + phatj - 3hatk + hatj - 2hatj +5hatk - 5hati + hatj + rhatk = barO`
`=> (2+q-5)hati + (p-2+1)hatj + (-3+5+r)hatk = 0hati + 0hatj + 0hatk`
by equality of vectors
2 + q - 5 = 0 ⇒ q = 3
p - 2 + 1 = 0 ⇒ p = 1
-3 + 5 + r = 0 ⇒ r = -2
∴ p = 1, q = 3 and r = -2
APPEARS IN
संबंधित प्रश्न
By vector method prove that the medians of a triangle are concurrent.
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)`
Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
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 \[{AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)\]
If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.
Let `A (bara)` and `B (barb)` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r = (mbarb + nbara)/(m + n) `
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.
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.
If two of the vertices of a triangle are A (3, 1, 4) and B(− 4, 5, −3) and the centroid of the triangle is at G (−1, 2, 1), then find the coordinates of the third vertex C of the triangle.
Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)
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"|.`
Find the volume of a parallelopiped whose coterimus edges are represented by the vectors `hat"i" + hat"k", hat"i" + hat"k", hat"i" + hat"j"`. Also find volume of tetrahedron having these coterminus edges.
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 G(a, 2, −1) is the centroid of the triangle with vertices P(1, 2, 3), Q(3, b, −4) and R(5, 1, c) then find the values of a, b and c
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 the angle bisectors of a triangle are concurrent
Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)
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 = ____________.
In a triangle ABC, if `1/(a + c) + 1/(b + c) = 3/(a + b + c)` then angle C is equal to ______
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 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 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 `3bar"a" + 5bar"b" = 8bar"c"`, then A divides BC in tbe ratio ______.
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"` = ______.
If D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC and 'O' is any point, then, `|vec(AD) + vec(BE) + vec(CF)|`, is ______.
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 ______.
ΔABC has vertices at A = (2, 3, 5), B = (–1, 3, 2) and C = (λ, 5, µ). If the median through A is equally inclined to the axes, then the values of λ and µ respectively are ______.
If G(g), H(h) and (p) are centroid orthocentre and circumcentre of a triangle and xp + yh + zg = 0, then (x, y, z) 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 vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by ______.
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.
Find the ratio in which the point C divides segment AB, if `5bara + 4barb - 9barc = bar0`
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)`.
AB and CD are two chords of a circle intersecting at right angles to each other at P. If R is the centre of the circle, prove that:
`bar(PA) + bar(PB) + bar(PC) + bar(PD) = 2bar(PR)`
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 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 + 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.`
