Three vectors A, B and C add up to zero. Find which is false.
A. (A×B) × C is not zero unless B, C are parallel
B. (A×B).C is not zero unless B, C are parallel
C. If A, B, C define a plane, (A×B) ×C is in that plane
D. (A×B).C=|A||B||C|→ C2=A2+B2
We have to identify false statement from above
It is given that
Therefore taking cross product on both side
Now we know that when vectors are parallel then their cross product is zero
Taking post Cross Product on both side with C
Now this could only zero when B and C are parallel to each other as,
only when that’s when B and C are parallel
Therefore statement A is true.
Now taking previous equation
Taking dot product with C on both side
Now this could be zero on two conditions first is that B and C are parallel but it could be zero without C being parallel to B. As when we will take cross product of B and C, then vector perpendicular to both B and C, say vector K. And by taking dot product of K and C it will also be zero as angle between them will always be 90.Therefore B is false
Now if vector triple product of A and B and C, then vector will always lie on place which will formed by A, B and C. This could be visualized by understanding that will always lie in a single plane forming sides of triangle.
K will be perpendicular to plane containing A and B.
And taking cross product with C (which is also lying on same plane as that of A and B) will give a vector which is perpendicular to C but will be lying on same plane as that of A,B and C. Therefore statement C is true.
It is given in last option that therefore angle between vector A and vector B is 90 and we know that
, therefore form a triangle with angle between A and B equal to 90, therefore it is right angled triangle. Hence option D is also true.
Rate this question :
Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates A = Axî + Ayĵ where î and ĵ are unit vector along x and y directions, respectively and Ax and Ay are corresponding components of (Fig. 4.9). Motion can also be studied by expressing vectors in circular polar co-ordinates as A = Ar + Aɵ where = = cosθ î + sin θ ĵ and = − sin θ î +cos θ ĵ are unit vectors along direction in which ‘r’ and ‘θ ’ are increasing.
(a) Express î and ĵ in terms of and .
(b) Show that both ř and θ are unit vectors and are perpendicular to each other.
(c) Show that (ř) = ω where and = −ωř
(d) For a particle moving along a spiral given by r= aθ , where a = 1 (unit), find dimensions of ‘a’.
(e) Find velocity and acceleration in polar vector representation for particle moving along spiral described in (d) above.
Physics - Exemplar
A man wants to reach from A to the opposite corner of the square C (Fig. 4.10). The sides of the square are 100 m. A central square of 50m × 50m is filled with sand. Outside this square, he can walk at a speed 1 m/s. In the central square, he can walk only at a speed of v m/s (v < 1). What is smallest value of v for which he can reach faster via a straight path through the sand than any path in the square outside the sand?
Physics - Exemplar
If |A|= 2 and |B| = 4, then match the relations in column I with the angle θ between A and B in column II
If |A| = 2 and |B| = 4, then match the relations in column I with the angle θ between A B and in column II.
A, B and C are three non-collinear, non-co-planar vectors. What can you say about direction of A × (B × C)?Physics - Exemplar
Consider the quantities, pressure, power, energy, impulse, gravitational potential, electrical charge, temperature, area. Out of these, the only vector quantities arePhysics - Exemplar