Advertisements
Advertisements
प्रश्न
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 \[\vec{a} + \vec{b} + \vec{c} .\]
उत्तर
Let, ABC be a given equilateral triangle and its vertices are A(\[\overrightarrow {a}\]), B([\overrightarrow {b}\]) , c(\[\overrightarrow {c}\]). Also,O(\[\overrightarrow {O}\]) be the orthocentre of triangle ABC.
We know that centroid and orthocentre of equilateral triangle coincide at one point.
\[\text{ Orthocentre of }\bigtriangleup ABC = \overrightarrow{O} \]
\[ \Rightarrow\text{ Centroid }\bigtriangleup ABC = \overrightarrow{O} \]
\[ \Rightarrow \frac{\overrightarrow{a} \ + \overrightarrow{b} + \overrightarrow{c}}{3} = \overrightarrow{o} \]
\[ \therefore \overrightarrow {a}+ \overrightarrow {b} + \overrightarrow {c} = \overrightarrow{0}\]
APPEARS IN
संबंधित प्रश्न
If `veca=xhati+2hatj-zhatk and vecb=3hati-yhatj+hatk` are two equal vectors ,then write the value of x+y+z
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] represent the sides of a triangle taken in order, then write the value of \[\vec{a} + \vec{b} + \vec{c} .\]
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.
In a regular hexagon ABCDEF, A \[\vec{B}\] = a, B \[\vec{C}\] = \[\overrightarrow{b}\text{ and }\overrightarrow{CD} = \vec{c}\].
Then, \[\overrightarrow{AE}\] =
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] and \[\vec{d}\] are the position vectors of points A, B, C, D such that no three of them are collinear and \[\vec{a} + \vec{c} = \vec{b} + \vec{d} ,\] then ABCD is a
Find the components along the coordinate axes of the position vector of the following point :
S(4, –3)
In the given figure express `bar"c"` and `bar"d"` in terms of `bar"a"` and `bar"b"`.
Find a vector in the direction of `bara = hati - 2hatj` that has magnitude 7 units.
Find the coordinates of the point which is located three units behind the YZ-plane, four units to the right of XZ-plane, and five units above the XY-plane.
If `|bar"a"| = |bar"b"| = 1, bar"a".bar"b" = 0, bar"a" + bar"b" + bar"c" = bar"0", "find" |bar"c"|`.
Find the lengths of the sides of the triangle and also determine the type of a triangle:
A(2, -1, 0), B(4, 1, 1), C(4, -5, 4)
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.
If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that `bar"QP" = 3bar"QG"`.
Dot product of a vector with vectors `3hat"i" - 5hat"k", 2hat"i" + 7hat"j" and hat"i" + hat"j" + hat"k"` are respectively -1, 6 and 5. Find the vector.
Show that no line in space can make angles `pi/6` and `pi/4` with X-axis and Y-axis.
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" xx bar"c")`
If `bar"a", bar"b", bar"c"` are three non-coplanar vectors show that `(bar"a".(bar"b" xx bar"c"))/((bar"c" xx bar"a").bar"b") + (bar"b".(bar"a" xx bar"c"))/((bar"c" xx bar"a").bar"b") = 0`
For any vector `overlinex` the value of `(overlinex xx hati)^2 + (overlinex xx hatj)^2 + (overlinex xx hatk)^2` is equal to ______
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 = ______.
lf `overlinea`, `overlineb` and `overlinec` are unit vectors such that `overlinea + overlineb + overlinec = overline0` and angle between `overlinea` and `overlineb` is `pi/3`, then `|overlinea xx overlineb| + |overlineb xx overlinec| + |overlinec xx overlinea|` = ______
Find a vector `vec"r"` of magnitude `3sqrt(2)` units which makes an angle of `pi/4` and `pi/2` with y and z-axes, respectively.
If `|vec"a"|` = 3 and –1 ≤ k ≤ 2, then `|"k"vec"a"|` lies in the interval ______.
If `vec"r" * vec"a" = 0, vec"r" * vec"b" = 0` and `vec"r" * vec"c" = 0` for some non-zero vector `vec"r"`, then the value of `vec"a" * (vec"b" xx vec"c")` is ______.
If `vec"a"` is any non-zero vector, then `(vec"a" .hat"i")hat"i" + (vec"a".hat"j")hat"j" + (vec"a".hat"k")hat"k"` equals ______.
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.
Equal
Let `bara, barb` and `barc` be three vectors, then `bara xx (barb xx barc) = (bara xx barb) xx barc` if
Four vectors `veca, vecb, vecc` and `vecx` satisfy the relation `(veca.vecx)vecb = vecc + vecx` where `vecb * veca` ≠ 1. The value of `vecx` in terms of `veca, vecb` and `vecc` is equal to
For given vectors, `veca = 2hati - hatj + 2hatk` and `vecb = - hati + hatj - hatk` find the unit vector in the direction of the vector `veca + vecb`.
Unit vector along `vec(PQ)`, where coordinates of P and Q respectively are (2, 1, – 1) and (4, 4, – 7), is ______.
Check whether the vectors `2 hati + 2 hatj + 3 hatk, -3 hati + 3 hatj + 2 hatk "and" 3 hati + 4 hatk` from a triangle or not.
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 mid-point of PR is M. Find following vectors in terms of `bara` and `barb`.
- `bar(PR)`
- `bar(PM)`
- `bar(QM)`
In the triangle PQR, `bar(PQ)=2bara` and `bar(QR)=2barb`. The mid-point of PR is M. Find following vectors in terms of `bara and barb`.
(i) `bar(PR)` (ii) `bar(PM)` (iii) `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.
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)`
