Advertisements
Advertisements
Question
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
Solution
Let `bar"p", bar"q", bar"r"` be the position vectors of points P, Q, R respectively of ∆PQR and g be the position vector of its centroid G.
∴ `bar"p" = hat"i" + 2hat"j" + 3hat"k", bar"q" = 3hat"i" + "b"hat"j" - 4hat"k", bar"r" = 5hat"i" + hat"j" + "c"hat"k"` and `bar"g" = "a"hat"i" + 2hat"j" - hat"k"`
∴ By using centroid formula,
`bar"g" = (bar"p" + bar"q" + bar"r")/3`
∴ `3bar"g" = bar"p" + bar"q" + "r"`
∴ `3("a"hat"i" + 2hat"j" - hat"k") = (hat"i" + 2hat"j" + 3hat"k") + (3hat"i" + "b"hat"j" - 4hat"k") + (5hat"i" + hat"j" + "c"hat"k")`
∴ `3"a"hat"i" + 6hat"j" - 3hat"k" = (1 + 3 + 5)hat"i" + (2 + "b" + 1)hat"j" + (3 - 4 + "c")hat"k"`
∴ `3"a"hat"i" + 6hat"j" - 3hat"k" = 9hat"i" + ("b" + 3)hat"j" + ("c" - 1)hat"k"`
∴ By equality of vectors, we get
3a = 9, 6 = b + 3 and −3 = c − 1
∴ a = 3, b = 3 and c = −2
APPEARS IN
RELATED QUESTIONS
By vector method prove that the medians of a triangle are concurrent.
Find the coordinate of the point P where the line through A(3, –4, –5) and B(2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0).
Also, find the ratio in which P divides the line segment AB.
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.
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 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.
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.
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 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.
Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half of the sum of the lengths of the parallel sides.
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.
If the centroid of a tetrahedron OABC is (1, 2, - 1) where A(a, 2, 3), B(1, b, 2), C(2, 1, c), find the distance of P(a, b, c) from origin.
Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)
If D, E, F are the midpoints of the sides BC, CA, AB of a triangle ABC, prove that `bar"AD" + bar"BE" + bar"CF" = bar0`.
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 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 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 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.
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 ______
The image of the point (1, 6, 3) in the line `x/1 = (y - 1)/2 = (z - 2)/3` is ______
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 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 ______.
Let `square`PQRS be a quadrilateral. If M and N are midpoints of the sides PQ and RS respectively then `bar"PS" + bar"OR"` = ______.
Find the unit vector in the diret:tion of the vector `veca = hati + hatj + 2hatk`
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 ______.
M and N are the mid-points of the diagonals AC and BD respectively of quadrilateral ABCD, then AB + AD + CB + CD is equal to ______.
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 ______.
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".`
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` .
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)`.
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 `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.`
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`.
The position vectors of points A and B are 6`bara` + 2`barb` 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 - b`.
