Advertisements
Advertisements
Question
If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that `bar"QP" = 3bar"QG"`.
Solution
Let `bar"p" and bar"g"` be the position vectors of P and G w.r.t. the circumcentre Q.
i.e. `bar"QR" = bar"p" and bar"QG" = bar"g"`
We know that Q, G, P are collinear and G divides segment QP internally in the ratio 1 : 2.
∴ by section formula for internal division,
`bar"g" = (1.bar"p" + 2bar"q")/(1 + 2) = bar"p"/3` .....`[∵ bar"q" = 0]`
∴ `bar"p" = 3bar"g"`
∴ `bar"QP" = 3bar"QG".`
APPEARS IN
RELATED QUESTIONS
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are position vectors of the vertices A, B and C respectively, of a triangle ABC, write the value of \[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} .\]
If D, E, F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC, write the value of \[\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} .\]
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then write the value of \[\vec{a} + \vec{b} + \vec{c} .\]
Write a unit vector in the direction of \[\overrightarrow{b} = 2 \hat{i} + \hat{j} + 2 \hat{k}\].
Find a unit vector in the direction of \[\overrightarrow{a} = 2 \hat{i} - 3 \hat{j} + 6 \hat{k}\].
Find the value of 'p' for which the vectors \[3 \hat{i} + 2 \hat{j} + 9 \hat{k}\] and \[\hat{i} - 2p \hat{j} + 3 \hat{k}\] are parallel.
If \[\vec{a} , \vec{b}\] are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is
Forces 3 O \[\vec{A}\], 5 O \[\vec{B}\] act along OA and OB. If their resultant passes through C on AB, then
If G is the intersection of diagonals of a parallelogram ABCD and O is any point, then \[O \vec{A} + O \vec{B} + O \vec{C} + O \vec{D} =\]
The vector equation of the plane passing through \[\vec{a} , \vec{b} , \vec{c} ,\text{ is }\vec{r} = \alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} ,\] provided that
Let G be the centroid of ∆ ABC. If \[\overrightarrow{AB} = \vec{a,} \overrightarrow{AC} = \vec{b,}\] then the bisector \[\overrightarrow{AG} ,\] in terms of \[\vec{a}\text{ and }\vec{b}\] is
Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Hence find whether the plane thus obtained contains the line \[\frac{x + 2}{5} = \frac{y - 3}{4} = \frac{z}{5}\] or not.
Find the value of λ for which the four points with position vectors `6hat"i" - 7hat"j", 16hat"i" - 19hat"j" - 4hat"k" , lambdahat"j" - 6hat"k" "and" 2hat"i" - 5hat"j" + 10hat"k"` are coplanar.
If` vec"a" = 2hat"i" + 3hat"j" + + hat"k", vec"b" = hat"i" - 2hat"j" + hat"k" "and" vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`
Find the area of the traingle with vertices (1, 1, 0), (1, 0, 1) and (0, 1, 1).
If `|bar"a"| = |bar"b"| = 1, bar"a".bar"b" = 0, bar"a" + bar"b" + bar"c" = bar"0", "find" |bar"c"|`.
Two sides of a parallelogram are `3hat"i" + 4hat"j" - 5hat"k"` and `-2hat"j" + 7hat"k"`. Find the unit vectors parallel to the diagonals.
Find the angle between the lines whose direction cosines are given by the equations 6mn - 2nl + 5lm = 0, 3l + m + 5n = 0.
Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0) and (0, 0, c). What is the area of the triangle with these vertices?
State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:
`(bar"a".bar"b").bar"c"`
State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:
`(bar"a".bar"b")bar"c"`
The XZ plane divides the line segment joining the points (3, 2, b) and (a, -4, 3) in the ratio ______.
The vector eqliation of line 2x - 2 = 3y + 1 = 6z - 2 is
a and b are non-collinear vectors. If c = (x - 2)a + b and d = (2x + 1)a - b are collinear vectors, then the value of x = ______.
If `|vec"a"|` = 8, `|vec"b"|` = 3 and `|vec"a" xx vec"b"|` = 12, then value of `vec"a" * vec"b"` is ______.
The vector `vec"a" + vec"b"` bisects the angle between the non-collinear vectors `vec"a"` and `vec"b"` if ______.
The values of k for which `|"k"vec"a"| < |vec"a"|` and `"k"vec"a" + 1/2 vec"a"` is parallel to `vec"a"` holds true are ______.
Classify the following measures as scalar and vector.
2 meters north-west
Classify the following measures as scalar and vector.
40 watt
In Figure, identify the following vector.
Collinear but not equal
If `veca ≠ vec(0), veca.vecb = veca.vecc, veca xx vecb = veca xx vecc`, then show that `vecb = vecc`.
Find the value of λ for which the points (6, – 1, 2), (8, – 7, λ) and (5, 2, 4) are collinear.
Check whether the vectors `2hati + 2hatj + 3hatk, -3hati + 3hatj + 2hatk` and `3hati + 4hatk` form a triangle or not.
In the triangle PQR, `bb(bar(PQ) = 2 bara)` and `bb(bar(QR) = 2 barb)`. The mid-point of PR is M. Find the following vectors in terms of `bb(bara and barb)`.
- `bar(PR)`
- `bar(PM)`
- `bar(QM)`
In the triangle PQR, `bar"PQ" = 2 bar" a" and bar"QR" = 2 bar"b"`. The midpoint of PR is M. Find the following vectors in terms of `bar"a"` and `bar"b"`:
(i) `bar"PR"` (ii) `bar"PM"` (iii) `bar"QM"`
Check whether the vectors `2hati + 2hatj + 3hatk, -3hati + 3hatj + 2hatk and 3hati + 4hatk` form a triangle or not.
In the triangle PQR, `bar(PQ)` = 2`bara` and `bar(QR)` = 2`barb`. The midpoint of PR is M. Find the following vectors in terms of `bara` and `barb`.
(i) `bar(PR)` (ii) `bar(PM)` (iii) `bar(QM)`
