Advertisements
Advertisements
प्रश्न
Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.
उत्तर
Let `hat"OP"` and `hat"OQ"` be unit vectors making angles A and B, respectively, with positive direction of x-axis.
Then ∠QOP = A – B .....[From the figure]
We know `hat"OP" = vec"OM" + vec"MP"`
= `hat"i" cos "A" + hat"j" sin "A"` and `hat"OQ" = vec"ON" + vec"NQ" = hat"i" cos"B" + hat"j" cos"B"`.
By definition `hat"OP" * hat"QO" = |hat"OP"| |hat"OQ"| cos("A" - "B")`
= `cos("A" - "B")` ......(1) `(because hat"OP"|= 1 = |hat"OQ"|)`
In terms of components, we have
`hat"QP" * hat"OQ" = (hat"i" cos"A" + hat"j" sin "A")*(hat"i" cos"B" + hat"j" sin"B")`
= cosA cosB + sinA sinB .....(2)
From (1) and (2), we get
cos(A – B) = cosA cosB + sinA sinB.
APPEARS IN
संबंधित प्रश्न
Write the position vector of a point dividing the line segment joining points A and B with position vectors \[\vec{a}\] and \[\vec{b}\] externally in the ratio 1 : 4, where \[\overrightarrow{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \text{ and }\overrightarrow{b} = - \hat{i} + \hat{j} + \hat{k} .\]
Write a unit vector in the direction of \[\overrightarrow{b} = 2 \hat{i} + \hat{j} + 2 \hat{k}\].
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.
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
Find the components along the coordinate axes of the position vector of the following point :
P(3, 2)
Find the position vector of the mid-point of the vector joining the points
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"`.
Find a vector in the direction of `bara = hati - 2hatj` that has magnitude 7 units.
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:
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
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 component form of `bar"a"` if it lies in YZ-plane makes 60° with positive Y-axis and `|bar"a"| = 4`.
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.
ABCD is a parallelogram. E, F are the midpoints of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB.
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" xx bar"c")`
The XZ plane divides the line segment joining the points (3, 2, b) and (a, -4, 3) in the ratio ______.
lf `overlinea` and `overlineb` be two unit vectors and θ is the angle between them, then `|overlinea - overlineb|` is equal to ______
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 ______
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 `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"` and `vec"b"` are adjacent sides of a rhombus, then `vec"a" * vec"b"` = 0
Classify the following measures as scalar and vector.
10-19 coulomb
Classify the following as scalar and vector quantity.
Force
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
Check whether the vectors `2hati + 2hatj + 3hatk, - 3hati + 3hatj +2 hatk and 3hati + 4hatk` from 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 + 2 hatj + 3hatk, - 3hati + 3hatj + 2hatk and 3hati + 4hatk` From 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)`
