Is it possible that the resultant of two vectors be smaller than the smaller of the two vectors? If so, under what condition?

Yes, if the angle between the two vectors is more than 90^o but less than 270⁰. (cosΘ is negative)

Define vector product of two vectors. Also prove using vector method. sin(A+B)=sinAcosB+cosAsinB

If a⃗= (3,-1,-4), b⃗=(-2,4,-3) and c⃗=(-5,7,-1) findI)                   Unit vector along a⃗-2b⃗+c⃗ II)                The direction cosines of the line represented by the vector a⃗-2b⃗+c⃗

Solution

Given vectors are

a= (3,-1,-4), b=(-2,4,-3) and c=(-5,7,-1)

a-2b+c = (3,-1,-4) -2(-2,4,-3)+(-5,7,-1)

             = (3,-1,-4) -(-4,8,-6)+(-5,7,-1)

             ={3-(-4)+(-5) , -1-8+7, -4-(-6)+(-1)}

             = (2,-2, 1)

I)...

Proof Parallelogram  Law

Consider two vectors P and Q with an angle θ between them. The sum of vectors P and Q is given by the vector R, the resultant sum vector using the parallelogram law of vector addition. If the resultant vector R makes an angle ϕ with the vector P, then the formulas for its magnitude and direction are:

• |R| = (P² + Q² + 2PQ cos θ)
• β = tan^-1[(Q sin θ)/(P + Q cos θ)]

Let us first see the...

If A,B,C and D are any four points so that,vector AB.CD + vector BC.AD + vector CA.BD=0.

The resultant of two equal forces may have a magnitude equal toone of the forces. At what angle between the two equal forces,this is possible? Justify your answer.

Let the forces be F1 and F2 at an angle A and their resultant be F. According to question,

| F1 | = | F2 | = | F |

We know,

F^2 = F1^2 + F2^2 + 2 F1 F2 cos A

or, F^2 = F^2 + F^2 + 2 F F cos A

or, F^2 - 2F^2 = 2 F^2 cos A

or, - F^2 = 2 F^2 cos A

or, -1 = 2 cos A

or, cos A = -1/2

or, cos A = cos 120

So, A = 120 degrees.

Hence, when two forces of equal magnitude act at an angle of 120 degrees, the resultant is equal to one of the forces.