If Vector A = 3icap + 4jcap and B= 7icap + 24jcap the the vector having the same magnitude as B and same direction as A is

Question

If Vector A = 3icap + 4jcap and B= 7icap + 24jcap the the vector having the same magnitude as B and same direction as A is 

Answer 1

Notice that both of these vectors lie in a 2-space plane (i,j) not a 3-space so the solution is a bit easier - actually the same as when you did algebra to find lines in an (x,y) plane.  And remember that the unit vector (i,j,k) is for the (x,y,z) directions so these vectors do indeed lie in the (x,y) plane.

 

 

The magnitude of vector B is found by the Pythagorean Theorem.

   R = √(24^2+7^2)

 

The direction of vector A can be found as tan^1 = 4/3.

Or, easier I think, is to remember that a perpendicular has a slope of the negative reciprocal to the other and parallel has the same slope.

So the slope of A is 4/3.

 

Write two equations √(I^2 + J^2) = R = √(24^2+7^2)

and either  tan^1 (24/7) = tan^1 (4/3)

or (using slopes)   J/I = 4/3

 

Then solve the 2 simultaneous equations.

 

There might be an easier rule of thumb about how to do this but it escapes me at the moment.

 

BTW - this is the First and Only time I have ever heard the unit vector called i cap + j cap + k cap.

The usual and more correct verbalization of the vector is HAT;  i hat + j hat + k hat

(It is also quicker to say phonetically, and if you use cap, j cap k cap  becomes a bit of a tongue twister if said quickly.)