Advertisements
Advertisements
प्रश्न
The points A, B, C have position vectors `bar"a", bar"b" and bar"c"` respectively. The point P is the midpoint of AB. Find the vector `bar"PC"` in terms of `bar"a", bar"b", bar"c"`.
उत्तर
P is the mid-point of AB.
∴ `bar"p" = (bar"a" + bar"b")/2, "where" bar"p"` is the position vector of P.
Now, `bar"PC" = bar"c" - bar"p" = bar"c" - 1/2(bar"a" + bar"b")`
`= -1/2(bar"a" + bar"b") + bar"c"`
`= - 1/2 bar"a" - 1/2 bar"b" + bar"c"`
APPEARS IN
संबंधित प्रश्न
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)`
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.
In a triangle OAB,\[\angle\]AOB = 90º. If P and Q are points of trisection of AB, prove that \[{OP}^2 + {OQ}^2 = \frac{5}{9} {AB}^2\]
(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.
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.
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 the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.
Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.
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)
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`.
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.
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
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
Prove that medians 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
Let G be the centroid of a Δ ABC and O be any other point in that plane, then OA + OB + OC + CG = ?
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 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 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 ______.
What is the midpoint of the vector joining the point P(2, 3, 4) and Q(4, 1, –2)?
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 ______.
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` .
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)`.
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 `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`.
