Advertisements
Advertisements
प्रश्न
If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is ______.
विकल्प
`sqrt(2)`
`sqrt(3)`
1
2
उत्तर
If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is `underline sqrt(3)`.
Explanation:
Step 1: Given data
Let `vec a` and `vec b` be two unit vectors whose sum is also a unit vector `vec c`.
`vec a + vec b = vec c`
Also, `|vec a| = |vec b| = |vec c| = 1`
Step 2: Calculating the magnitude of the sum of the two vectors
Now, the magnitude of the sum of `vec a and vec b` is
`|vec a + vec b|^2 = |vec a|^2 + |vec b|^2 + 2 ("ab cos" theta)`
1 = 1 + 1 + 2(ab cos θ)
(ab cos θ) = `-1/2`
Step 3: Calculating the magnitude of the difference between the two vectors
The magnitude of their difference is given by
`|vec a - vec b|^2 = |vec a|^2 + |vec b|^2 - 2("ab cos" theta)`
`1 + 1 - 2 xx (-1/2)` = 3
`|vec a - vec b|^2 = 3` ...[Square root on both sides]
`|vec a - vec b| = sqrt3`
Therefore, the magnitude of the difference between the two vectors is `sqrt3`
APPEARS IN
संबंधित प्रश्न
If \[\overrightarrow{a}\] and \[\overrightarrow{b}\] denote the position vectors of points A and B respectively and C is a point on AB such that 3AC = 2AB, then write the position vector of C.
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 \[\overrightarrow{a} = \hat{i} + \hat{j} , \vec{b} = \hat{j} + \hat{k} \text{ and }\vec{c} = \hat{k} + \hat{i} ,\] write unit vectors parallel to \[\overrightarrow{a} + \overrightarrow{b} - 2 \overrightarrow{c} .\]
Write the position vector of a point dividing the line segment joining points having position vectors \[\hat{i} + \hat{j} - 2 \hat{k} \text{ and }2 \hat{i} - \hat{j} + 3 \hat{k}\] externally in the ratio 2:3.
If \[\overrightarrow{a} = \hat{i} + \hat{j} , \overrightarrow{b} = \hat{j} + \hat{k} , \overrightarrow{c} = \hat{k} + \hat{i}\], find the unit vector in the direction of \[\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}\].
Write a unit vector in the direction of \[\overrightarrow{a} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} .\]
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 \[\overrightarrow{a} + \overrightarrow{b}\].
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.
Write the position vector of the point which divides the join of points with position vectors \[3 \overrightarrow{a} - 2 \overrightarrow{b}\text{ and }2 \overrightarrow{a} + 3 \overrightarrow{b}\] in the ratio 2 : 1.
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 \[\vec{a} + \vec{b} + \vec{c} =\]
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
The position vectors of the points A, B, C are \[2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k}\text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}\] respectively.
These points
ABCD is a parallelogram with AC and BD as diagonals.
Then, \[\overrightarrow{AC} - \overrightarrow{BD} =\]
Find the components along the coordinate axes of the position vector of the following point :
S(4, –3)
If `veca` and `vecb` are non- collinear vectors, find the value of x such that the vectors `barα = (x - 2)veca + vecb` and `barβ = (3+2x)bara - 2barb` are collinear.
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"]`
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.
Find the coordinates of the point which is located in the YZ-plane, one unit to the right of the XZ- plane, and six units above the XY-plane.
Select the correct option from the given alternatives:
The volume of tetrahedron whose vectices are (1,-6,10), (-1, -3, 7), (5, -1, λ) and (7, -4, 7) is 11 cu units, then the value of λ is
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 ______
If two sides of a triangle are `hat"i" + 2hat"j" and hat"i" + hat"k"`, find the length of the third side.
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).
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"|).`
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`.
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.
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" xx bar"b").(bar"c"xxbar"d")`
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"`
For any vector `overlinex` the value of `(overlinex xx hati)^2 + (overlinex xx hatj)^2 + (overlinex xx hatk)^2` is equal to ______
If the vectors `xhat"i" - 3hat"j" + 7hat"k" and hat"i" + "y"hat"j" - "z"hat"k"` are collinear then the value of `"xy"^2/"z"` is equal.
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 the points (–1, –1, 2), (2, m, 5) and (3,11, 6) are collinear, find the value of m.
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"|` = 8, `|vec"b"|` = 3 and `|vec"a" xx vec"b"|` = 12, then value of `vec"a" * vec"b"` is ______.
The unit vector perpendicular to the vectors `hat"i" - hat"j"` and `hat"i" + hat"j"` forming a right handed system is ______.
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"` and `vec"b"` are adjacent sides of a rhombus, then `vec"a" * vec"b"` = 0
Classify the following as scalar and vector quantity.
Distance
In Figure, identify the following vector.
Equal
In Figure, identify the following vector.
Collinear but not equal
Let `veca, vecb` and `vecc` be three unit vectors such that `veca xx (vecb xx vecc) = sqrt(3)/2 (vecb + vecc)`. If `vecb` is not parallel to `vecc`, then the angle between `veca` and `vecc` is
Check whether the vectors `2hati + 2hatj + 3hatk, -3hati + 3hatj + 2hatk` and `3hati + 4hatk` form a triangle or not.
If `|veca| = 3, |vecb| = sqrt(2)/3` and `veca xx vecb` is a unit vector then the angle between `veca` and `vecb` will be ______.
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)`
