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Degree of the differential equation `sinx + cos(dy/dx)` = y2 is ______.
Concept: Order and Degree of a Differential Equation
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
Concept: Methods of Solving First Order, First Degree Differential Equations >> Homogeneous Differential Equations
Find the general solution of the differential equation:
`(x^2 + 1) dy/dx + 2xy = sqrt(x^2 + 4)`
Concept: Methods of Solving First Order, First Degree Differential Equations >> Linear Differential Equations
The sum of the order and the degree of the differential equation `d/dx[(dy/dx)^3]` is ______.
Concept: Order and Degree of a Differential Equation
Prove that, for any three vector `veca,vecb,vecc [vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`
Concept: Scalar Triple Product of Vectors
Show that the points A, B, C with position vectors `2hati- hatj + hatk`, `hati - 3hatj - 5hatk` and `3hati - 4hatj - 4hatk` respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle
Concept: Introduction of Product of Two Vectors
If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, find the angle which `veca + vecb + vecc`make with `veca or vecb or vecc`
Concept: Magnitude and Direction of a Vector
Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).
Concept: Vectors Examples and Solutions
If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is `sqrt(3)`.
Concept: Magnitude and Direction of a Vector
Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk` and `vecb = 3hati - 2hatj + 2hatk` .
Concept: Product of Two Vectors >> Vector (Or Cross) Product of Two Vectors
Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.
Concept: Scalar Triple Product of Vectors
Let `veca` , `vecb` and `vecc` be three vectors such that `|veca| = 1,|vecb| = 2, |vecc| = 3.` If the projection of `vecb` along `veca` is equal to the projection of `vecc` along `veca`; and `vecb` , `vecc` are perpendicular to each other, then find `|3veca - 2vecb + 2vecc|`.
Concept: Product of Two Vectors >> Projection of a Vector on a Line
if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.
Concept: Basic Concepts of Vector Algebra
Projection of vector `2hati + 3hatj` on the vector `3hati - 2hatj` is ______.
Concept: Product of Two Vectors >> Projection of a Vector on a Line
A line l passes through point (– 1, 3, – 2) and is perpendicular to both the lines `x/1 = y/2 = z/3` and `(x + 2)/-3 = (y - 1)/2 = (z + 1)/5`. Find the vector equation of the line l. Hence, obtain its distance from the origin.
Concept: Basic Concepts of Vector Algebra
Two vectors `veca = a_1 hati + a_2 hatj + a_3 hatk` and `vecb = b_1 hati + b_2 hatj + b_3 hatk` are collinear if ______.
Concept: Components of Vector
The magnitude of the vector `6hati - 2hatj + 3hatk` is ______.
Concept: Magnitude and Direction of a Vector
Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line `(x+3)/3=(4-y)/5=(z+8)/6`
Concept: Equation of a Line in Space
Find the distance between the planes 2x - y + 2z = 5 and 5x - 2.5y + 5z = 20
Concept: Shortest Distance Between Two Lines
The x-coordinate of a point of the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its z-coordinate
Concept: Vector and Cartesian Equation of a Plane
