Advertisements
Advertisements
प्रश्न
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"|).`
उत्तर
Choose any point P on the angle bisector of ∠AOB. Draw PM parallel to OB.
∴ ∠OPM = ∠POM = ∠POB
Hence, OM = MP
∴ OM and MP is the same scalar multiple of unit vectors `hat"a" and hat"b"` along these directions,
where `hat"a" = bar"a"/|bar"a"| and hat"b" = bar"b"/|bar"b"|`
∴ `bar"OM" = lambdahat"a" and bar"MP" = lambdahat"b"`
∴ `bar"OP" = bar"OM" + bar"MP"`
`= lambdahat"a" + lambdahat"b"`
`= lambda(hat"a" + hat"b")`
Hence, the vector along angle bisector of ∠AOB is given by
`bar"d" = bar"OP" = lambda(bar"a"/|bar"a"| + bar"b"/|bar"b"|)`.
APPEARS IN
संबंधित प्रश्न
Write a unit vector making equal acute angles with the coordinates axes.
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 \[\overrightarrow{a} + \overrightarrow{b}\].
Find the position vector of the mid-point of the line segment AB, where A is the point (3, 4, −2) and B is the point (1, 2, 4).
If \[\left| \overrightarrow{a} \right| = 4\] and \[- 3 \leq \lambda \leq 2\], then write the range of \[\left| \lambda \vec{a} \right|\].
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 points A (60 \[\hat{i}\] + 3 \[\hat{j}\]), B (40 \[\hat{i}\] − 8 \[\hat{j}\]) and C (a \[\hat{i}\] − 52 \[\hat{j}\]) are collinear, then a is equal to
If three points A, B and C have position vectors \[\hat{i} + x \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k}\text{ and }y \hat{i} - 2 \hat{j} - 5 \hat{k}\] respectively are collinear, then (x, y) =
Find the components along the coordinate axes of the position vector of the following point :
P(3, 2)
Find the components along the coordinate axes of the position vector of the following point :
R(–11, –9)
Find a unit vector perpendicular to each of the vectors `veca + vecb "and" veca - vecb "where" veca = 3hati + 2hatj + 2hatk and vecb = i + 2hatj - 2hatk`
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.
Find the area of the traingle with vertices (1, 1, 0), (1, 0, 1) and (0, 1, 1).
If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is ______.
Select the correct option from the given alternatives:
Let a, b, c be distinct non-negative numbers. If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k" , hat"i" + hat"k" "and" "c"hat"i" + "c"hat"j" + "b"hat"k"` lie in a plane, then c is
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"`.
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 unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).
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".`
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")`
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.
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 = ______.
For any non zero vector, a, b, c a · ((b + c) × (a + b + c)] = ______.
For 0 < θ < π, if A = `[(costheta, -sintheta), (sintheta, costheta)]`, then ______
Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.
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 `hat"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `2vec"a" - vec"b"`
Classify the following as scalar and vector quantity.
Distance
Classify the following as scalar and vector quantity.
Work done
In Figure, identify the following vector.
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
Let `bara, barb` and `barc` be three vectors, then `bara xx (barb xx barc) = (bara xx barb) xx barc` if
Which of the following measures as vector?
If `veca` and `vecb` are two collinear vectors then which of the following are incorrect.
Check whether the vectors`2hati+2hatj+3hatk,-3hati+3hatj+2hatk and 3hati +4hatk` form a triangle or not.
In the triangle PQR, `bar(PQ) = 2bara and bar(QR) = 2barb`. The mid-point of PR is M. Find the 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.
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)`
