Advertisements
Advertisements
प्रश्न
Find the lengths of the sides of the triangle and also determine the type of a triangle:
L (3, -2, -3), M (7, 0, 1), N(1, 2, 1).
उत्तर
The position vectors bar"a", bar"b", bar"c" of the points L, M, N are
`bar"a" = 3hat"i" - 2hat"j" - 3hat"k", bar"b" = 7hat"i" + hat"k", bar"c" = hat"i" + 2hat"j" + hat"k"`
`bar"LM" = bar"b" - bar"a" = (7hat"i" + hat"k") - (3hat"i" - 2hat"j" - 3hat"k") = 4hat"i" + 2hat"j" + 4hat"k"`
`bar"MN" = bar"c" - bar"b" = (hat"i" + 2hat"j" + hat"k") - (7hat"i" + hat"k") = - 6hat"i" + 2hat"j"`
`bar"NL" = bar"a" - bar"c" = (3hat"i" - 2hat"j" - 3hat"k") - (hat"i" + 2hat"j" + hat"k")= 2hat"i" - 4hat"j" - 4hat"k"`
∴ l(LM) = `|bar"AB"| = sqrt(4^2 + 2^2 + 4^2) = sqrt(16 + 4 + 16) = sqrt36 = 6`
l(MN) = `|bar"MN"| = sqrt((- 6)^2 + 2^2) = sqrt(36 + 4) = sqrt40 = sqrt(10 xx 4) = 2sqrt10`
l(NL) = `|bar"NL"| = sqrt((2)^2 + (-4)^2 + (- 4)^2) = sqrt(4 + 16 + 16) = sqrt36 = 6`
l(LM) = 6, l(MN) = `2sqrt10`, l(NL) = 6
∴ Δ LMN is isosceles.
APPEARS IN
संबंधित प्रश्न
If \[\vec{a}\] and \[\vec{b}\] represent two adjacent sides of a parallelogram, then write vectors representing its diagonals.
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 \[\overrightarrow{a}\], \[\overrightarrow{b}\], \[\overrightarrow{c}\] are the position vectors of the vertices of a triangle, then write the position vector of its centroid.
If \[\overrightarrow{a} = \hat{i} + 2 \hat{j} , \vec{b} = \hat{j} + 2 \hat{k} ,\] write a unit vector along the vector \[3 \overrightarrow{a} - 2 \overrightarrow{b} .\]
Write a unit vector in the direction of \[\overrightarrow{a} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} .\]
Write a unit vector in the direction of \[\overrightarrow{b} = 2 \hat{i} + \hat{j} + 2 \hat{k}\].
For what value of 'a' the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k} \text{ and }a \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear?
Find a unit vector in the direction of the vector \[\overrightarrow{a} = 3 \hat{i} - 2 \hat{j} + 6 \hat{k}\].
If \[\overrightarrow{a} = x \hat{i} + 2 \hat{j} - z \hat{k}\text{ and }\overrightarrow{b} = 3 \hat{i} - y \hat{j} + \hat{k}\] are two equal vectors, then write the value of x + y + z.
ABCDEF is a regular hexagon. Show that `bar"AB" + bar"AC" + bar"AD" + bar"AE" + bar"AF" = 6bar"AO"`, where O is the centre of the hexagon.
Express the vector `bar"a" = 5hat"i" - 2hat"j" + 5hat"k"` as a sum of two vectors such that one is parallel to the vector `bar"b" = 3hat"i" + hat"k"` and other is perpendicular to `bar"b"`.
Show that the vector area of a triangle ABC, the position vectors of whose vertices are `bar"a", bar"b" and bar"c"` is `1/2[bar"a" xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a"]`.
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".bar"d")`
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"`
Find the volume of the parallelopiped spanned by the diagonals of the three faces of a cube of side a that meet at one vertex of the cube.
If `overline"u"` and `overline"v"` are unit vectors and θ is the acute angle between them, then `2overline"u" xx 3overline"v"` is a unit vector for ______
If A, B, C and D are (3, 7, 4), (5, -2, - 3), (- 4, 5, 6) and(1, 2, 3) respectively, then the volume of the parallelopiped with AB, AC and AD as the co-terminus edges, is ______ cubic units.
If the vectors `overlinea = 2hati - qhatj + 3hatk` and `overlineb = 4hati - 5hatj + 6hatk` are collinear, then the value of q is ______
If `|vec"a"|` = 8, `|vec"b"|` = 3 and `|vec"a" xx vec"b"|` = 12, then value of `vec"a" * vec"b"` is ______.
If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `6vec"b"`
If `vec"a", vec"b", vec"c"` determine the vertices of a triangle, show that `1/2[vec"b" xx vec"c" + vec"c" xx vec"a" + vec"a" xx vec"b"]` gives the vector area of the triangle. Hence deduce the condition that the three points `vec"a", vec"b", vec"c"` are collinear. Also find the unit vector normal to the plane of the triangle.
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 ______.
The formula `(vec"a" + vec"b")^2 = vec"a"^2 + vec"b"^2 + 2vec"a" xx vec"b"` is valid for non-zero vectors `vec"a"` and `vec"b"`
If `vec"a"` and `vec"b"` are adjacent sides of a rhombus, then `vec"a" * vec"b"` = 0
Classify the following measures as scalar and vector.
40 watt
Classify the following as scalar and vector quantity.
Force
If `veca ≠ vec(0), veca.vecb = veca.vecc, veca xx vecb = veca xx vecc`, then show that `vecb = vecc`.
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
Find `|vecx|`, if for a unit vector `veca, (vecx - veca) * (vecx + veca)` = 12
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.
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)`
If `hata` is unit vector and `(2vecx - 3hata)*(2vecx + 3hata)` = 91, find the value of `|vecx|`.
lf ΔABC is an equilateral triangle and length of each side is “a” units, then the value of `bar(AB)*bar(BC) + bar(BC)*bar(CA) + bar(CA)*bar(AB)` is ______.
Evaluate the following.
`int x^3/(sqrt1 + x^4) `dx
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"`
In the triangle PQR, `bar"PQ" = bar"2a", bar"QR" = bar"2b"`. 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"`.
Consider the following statements and choose the correct option:
Statement 1: If `veca` and `vecb` represents two adjacent sides of a parallelogram then the diagonals are represented by `veca + vecb` and `veca - vecb`.
Statement 2: If `veca` and `vecb` represents two diagonals of a parallelogram then the adjacent sides are represented by `2(veca + vecb)` and `2(veca - vecb)`.
Which of the following is correct?
