Advertisements
Advertisements
प्रश्न
If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin2 α + sin2 β + sin2 γ.
उत्तर
Suppose, a vector \[\overrightarrow{OP}\] makes an angle \[\alpha, \beta, \gamma\] with OX, OY, OZ respectively.
Then, direction cosines of the vector are given by \[l = \cos \alpha , m = \cos \beta \text{ and }n = \cos \gamma\]
Consider,
\[\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1 - \cos^2 \alpha + 1 - \cos^2 \beta + 1 - \cos^2 \gamma\]
\[ = 3 - \left( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \right)\]
\[ = 3 - \left( l^2 + m^2 + n^2 \right)\]
= 3 - 1 [∵ \[l^2 + m^2 + n^2 = 1\]]
= 2
APPEARS IN
संबंधित प्रश्न
If \[\vec{a}\] and \[\vec{b}\] are two non-collinear vectors such that \[x \vec{a} + y \vec{b} = \vec{0} ,\] then write the values of x and y.
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 \[\vec{a} , \vec{b}\] are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is
Find the components along the coordinate axes of the position vector of the following point :
Q(–5, 1)
Select the correct option from the given alternatives:
If l, m, n are direction cosines of a line then `"l"hat
"i" + "m"hat"j" + "n"hat"k"` is ______
Select the correct option from the given alternatives:
If `bar"a" "and" bar"b"` are unit vectors, then what is the angle between `bar"a"` and `bar"b"` for `sqrt3bar"a" - bar"b"` to be a unit vector?
Let `bara = hati - hatj, barb = hatj - hatk, barc = hatk - hati.` If `bard` is a unit vector such that `bara * bard = 0 = [(barb, barc, bard)]`, then `bard` equals ______.
In a parallelogram ABCD, diagonal vectors are `bar"AC" = 2hat"i" + 3hat"j" + 4hat"k" and bar"BD" = - 6hat"i" + 7hat"j" - 2hat"k"`, then find the adjacent side vectors `bar"AB" and bar"AD"`.
Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).
If `bar"OA" = bar"a" and bar"OB" = bar"b",` then show that the vector along the angle bisector of ∠AOB is given by `bar"d" = lambda(bar"a"/|bar"a"| + bar"b"/|bar"b"|).`
A point P with position vector `(- 14hat"i" + 39hat"j" + 28hat"k")/5` divides the line joining A (1, 6, 5) and B in the ratio 3 : 2, then find the point B.
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.
If `bar"a", bar"b", bar"c"` are unit vectors such that `bar"a" + bar"b" + bar"c" = bar0,` then find the value of `bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a".`
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"`.
Find the acute angle between the curves at their points of intersection, y = x2, y = x3.
Let bar"b" = 4hat"i" + 3hat"j" and bar"c" be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along bar"b" and bar"c" respectively.
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")bar"c"`
For any vectors `bar"a", bar"b", bar"c"` show that `(bar"a" + bar"b" + bar"c") xx bar"c" + (bar"a" + bar"b" + bar"c") xx bar"b" + (bar"b" - bar"c") xx bar"a" = 2bar"a" xx bar"c"`
The vector eqliation of line 2x - 2 = 3y + 1 = 6z - 2 is
The points A(- a, -b), B (0, 0), C(a, b) and D(a2 , ab) are ______.
lf `overlinea` and `overlineb` be two unit vectors and θ is the angle between them, then `|overlinea - overlineb|` is equal to ______
For any non zero vector, a, b, c a · ((b + c) × (a + b + c)] = ______.
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|` = ______
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"|` = 3 and –1 ≤ k ≤ 2, then `|"k"vec"a"|` lies in the interval ______.
Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + hat"k"` and `vec"b" = 2hat"j" + hat"k"`.
Find a unit vector in the direction of `vec"PQ"`, where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively
In Figure, identify the following vector.
Equal
If `veca ≠ vec(0), veca.vecb = veca.vecc, veca xx vecb = veca xx vecc`, then show that `vecb = vecc`.
`bara, barb` and `barc` are three vectors such that `veca + vecb + vecc` 20, `|bara| = 1, |barb| = 2` and `|barc| = 3`. Then `bara. barb + barb.barc + bar(c.a)` is equal to
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
Let the vectors `vec(a)` such `vec(b)` that `|veca|` = 3 and `|vecb| = sqrt(2)/3`, then `veca xx vecb` is a unit vector if the angle between `veca` and `vecb` is
If `hata` is unit vector and `(2vecx - 3hata)*(2vecx + 3hata)` = 91, find the value of `|vecx|`.
Evaluate the following.
`int x^3/(sqrt1 + x^4) `dx
Check whether the vectors `2 hati+2 hatj+3 hatk,-3 hati+3 hatj+2 hatk and 3 hati +4 hatk` form a triangle or not.
Check whether the vectors `2hati + 2hatj + 3hatk, -3hati + 3hatj + 2hatk and 3hati + 4hatk` form a triangle or not.
