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
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.
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 using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.
Prove that the diagonals of a rhombus are perpendicular bisectors of each other.
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.
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) `
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"`.
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.
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"|.`
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
Prove that medians of a triangle are concurrent
Prove that altitudes 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
Let G be the centroid of a Δ ABC and O be any other point in that plane, then OA + OB + OC + CG = ?
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"` = ______
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 ______
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)?
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 ______.
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 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`
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 `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 + 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`.
