Advertisements
Advertisements
Question
Find two unit vectors each of which makes equal angles with bar"u", bar"v" and bar"w" where bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v" = hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k".
Solution
Let `bar"r" = "x"hat"i" + "y"hat"j" + "z"hat"k"` be the unit vector which makes angle θ with each of the vectors
Then `|bar"r"| = 1`
Also, `bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v"= hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k"`
`|bar"u"| = sqrt(2^2 + 1^2 + (- 2)^2) = sqrt(4 + 1 + 4) = sqrt9 = 3`
`|bar"v"| = sqrt(1^2 + 2^2 + (- 2)^2) = sqrt(1 + 4 + 4) = sqrt9 = 3`
`|bar"w"| = sqrt(2^2 + (- 2)^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt9 = 3`
Angle between `bar"r" and bar"u"` is θ
∴ cos θ = `(bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(2hat"i" + hat"j" - 2hat"k"))/(1xx3)`
`= (2"x" + "y" - 2"z")/3` ....(1)
Also, the angle between `bar"r" and bar"v"` and between `bar"r" and bar"w"` is θ.
∴ cos θ = `(bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(hat"i" + 2hat"j" - 2hat"k"))/(1xx3)`
`= ("x" + 2"y" - 2"z")/3` ....(2)
and cos θ `= (bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(2hat"i" - 2hat"j" + hat"k"))/(1xx3)`
`= (2"x" - 2"y" + "z")/3` ....(3)
From (1) and (2), we get
`(2"x" + "y" - 2"z")/3 = ("x" + 2"y" - 2"z")/3`
∴ 2x + y - 2z = x + 2y - 2z
∴ x = y
From (2) and (3), we get
`("x" + 2"y" - 2"z")/3 = (2"x" - 2"y" + "z")/3`
∴ x + 2y - 2z = 2x - 2y + z
∴ 3y = 3z .....[∵ x = y]
∴ y = z
∴ x = y = z
∴ `bar"r" = "x"hat"i" + "y"hat"j" + "z"hat"k" = "x"hat"i" + "x"hat"j" + "x"hat"k"`
∴ `|bar"r"| = sqrt("x"^2 + "x"^2 + "x"^2) = 1`
∴ `"x"^2 + "x"^2 + "x"^2 = 1`
∴ `3"x"^2 = 1`
∴ `"x"^2 = 1/3`
∴ x = `- 1/sqrt3 `
∴ `bar"r" = +- 1/sqrt3 hat"i" +- 1/sqrt3 hat"j" +- 1/sqrt3hat"k" `
`= +- 1/sqrt3 (hat"i" + hat"j" + hat"k")`
Hence, the required unit vectors are `+- 1/sqrt3 (hat"i" + hat"j" + hat"k")`
APPEARS IN
RELATED QUESTIONS
If \[\overrightarrow{a} = \hat{i} + 2 \hat{j} - 3 \hat{k} \text{ and }\overrightarrow{b} = 2 \hat{i} + 4 \hat{j} + 9 \hat{k} ,\] find a unit vector parallel to \[\overrightarrow{a} + \overrightarrow{b}\].
Write a unit vector in the direction of \[\overrightarrow{b} = 2 \hat{i} + \hat{j} + 2 \hat{k}\].
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 \[\vec{a} , \vec{b} , \vec{c}\] are three non-zero vectors, no two of which are collinear and the vector \[\vec{a} + \vec{b}\] is collinear with \[\vec{c} , \vec{b} + \vec{c}\] is collinear with \[\vec{a} ,\] then \[\vec{a} + \vec{b} + \vec{c} =\]
In a regular hexagon ABCDEF, A \[\vec{B}\] = a, B \[\vec{C}\] = \[\overrightarrow{b}\text{ and }\overrightarrow{CD} = \vec{c}\].
Then, \[\overrightarrow{AE}\] =
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
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 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 components along the coordinate axes of the position vector of the following point :
R(–11, –9)
In Figure, which of the following is not true?
The vector `bar"a"` is directed due north and `|bar"a"|` = 24. The vector `bar"b"` is directed due west and `|bar"b"| = 7`. Find `|bar"a" + bar"b"|`.
OABCDE is a regular hexagon. The points A and B have position vectors `bar"a"` and `bar"b"` respectively referred to the origin O. Find, in terms of `bar"a"` and `bar"b"` the position vectors of C, D and E.
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.
If a parallelogram is constructed on the vectors `bar"a" = 3bar"p" - bar"q", bar"b" = bar"p" + 3bar"q" and |bar"p"| = |bar"q"| = 2` and angle between `bar"p" and bar"q"` is `pi/3,` and angle between lengths of the sides is `sqrt7 : sqrt13`.
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"`.
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")`
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.
For any non-zero vectors a and b, [b a × b a] = ?
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 ______
For any vector `overlinex` the value of `(overlinex xx hati)^2 + (overlinex xx hatj)^2 + (overlinex xx hatk)^2` is equal to ______
For any non zero vector, a, b, c a · ((b + c) × (a + b + c)] = ______.
If the vectors `overlinea = 2hati - qhatj + 3hatk` and `overlineb = 4hati - 5hatj + 6hatk` are collinear, then the value of q is ______
Find a vector of magnitude 11 in the direction opposite to that of `vec"PQ"` where P and Q are the points (1, 3, 2) and (–1, 0, 8), respectively.
The vector with initial point P (2, –3, 5) and terminal point Q(3, –4, 7) is ______.
If `veca` and `vecb` are unit vectors, then what is the angle between `veca` and `vecb` for `sqrt(3) veca - vecb` to be a unit vector?
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" = hat"i" + hat"j" + 2hat"k"` and `hat"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `2vec"a" - vec"b"`
Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.
If `|vec"a" + vec"b"| = |vec"a" - vec"b"|`, then the vectors `vec"a"` and `vec"b"` are orthogonal.
Classify the following measures as scalar and vector.
2 meters north-west
Classify the following measures as scalar and vector.
40 watt
Classify the following measures as scalar and vector.
10-19 coulomb
Classify the following as scalar and vector quantity.
Force
In Figure, identify the following vector.
Collinear but not equal
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
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)`
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.
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.
