Advertisements
Advertisements
प्रश्न
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.
उत्तर
The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).
∴ `vec(AB) = (5 - 1)hati + (0 + 2)hatj + (-2 + 8)hatk `
`= 4hati + 2hatj + 6hatk`
`vec(BC) = (11 - 5)hati + (3 - 0)hatj + (7 + 2)hatk`
`= 6hati + 3hatj + 9hatk`
`vec(AC) = (11 - 1)hati + (3 + 2)hatj + (7 + 8)hatk`
`= 10hati + 5hatj + 15hhatk`
`|vec(AB)| = sqrt(4^2 + 2^2 + 6^2) = sqrt(16 + 4 + 36) `
`= sqrt56 `
`= 2sqrt14`
`|vec(BC)| = sqrt(6^2 + 3^2 + 9^2) `
`= sqrt(36 + 9 + 81)`
` = sqrt126 `
`= 3sqrt14`
`|vec(AC)| = sqrt(10^2 + 5^2 + 15^2)`
` = sqrt(100 + 25 + 225) `
`= sqrt350 `
`= 5sqrt14`
∴ `|vec(AC)| = |vec(AB)| + |vec(BC)|`
Thus, the given points A, B, and C are collinear.
Now, let point B divide AC in the ratio `lambda: 1`. Then, we have:
`vec(OB) = (lambdavec(OC) + vec(OA))/((lambda + 1))`
`⇒ 5hati - 2hatk = (lambda(11hati + 3hatj + 7hatk) + (hati - 2hatj - 8hatk))/(lambda + 1)`
⇒ `(lambda + 1)(5hati - 2hatk) = 11lambdahati + 3lambdahatj + 7lambdahatk + hati - 2hatj - 8hatk`
`⇒ 5(lambda + 1)hati - 2(lambda + 1)hatk = (11lambda + 1)hati + (3lambda - 2)hatj + (7lambda - 8)hatk`
On equating the corresponding components, we get:
5(λ + 1) = 11λ + 1
⇒ 5λ + 5 = 11λ + 1
⇒ 6λ = 4
⇒ λ = `4/6 + 2/3`
Hence, point B divides AC in the ratio of 2: 3
APPEARS IN
संबंधित प्रश्न
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 that the diagonals of a rhombus are perpendicular bisectors of each other.
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 \[{AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)\]
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 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 a quadrilateral is a parallelogram if and only if its diagonals bisect each other.
Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)
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"`.
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`.
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 altitudes of a triangle are concurrent
Prove that the angle bisectors of a triangle are concurrent
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.
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 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 ______.
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 ______.
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 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 ______.
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 ______.
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)`
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 + 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`.
