Can a rectangular component of a vector be greater than the vector itself?

Question

Can a rectangular component of a vector be greater than the vector itself?

Answer 1

No.

Let V^2 = vx^2 + vy^2; where V is the vector's magnitude, vx is the X component of V, and vy is the Y component of it.  This is the Pythagorian equation and V is the hypoteneuse of the rectangle formed by vx and vy.

As you can see, none of the components vx and vy can exceed the magnitude V.  One or the other might equal V, in which case we'd have a zero value for the non-V component.  But that's as close as the components can get to the size of a vector.

BTW...I should add, this is true in Euclidean space, which, by definition, is where the Pythagorian equation is valid.  In other space, like spherical, this may or may not be true.

Answer 2

No, you don't need any more explanation than above answers  

Answer 3

I am a little rusty. You are talking about a vector on a Cartesian Coordinate graph right? I mean with the x and y vectors (or i and j) being the rectangular components correct?

No, a component can't be larger than the vector itself. The vector can only be the same magnitude of a component vector if all other component vectors are zero.

I'm not sure if it would qualify but if you were finding the product of several vectors than one of the vectors could be greater in magnitude than the product vector.  

Answer 4

Think about it. The rectangular components are two adjacent sides of a rectangle whose diagonal is the vector. The diagonal of a rectangle is never less than the length of either of two adjacent sides.

At the very most, one side can increase to approach the length of the diagonal, as the adjacent side decreases to approach zero. This happens when a vector of constant length rotates around the origin of an X-Y coordinate system. At zero, and at every integer multiple of pi radians thereafter, one of the rectangular components becomes zero when the other component is the length of the vector.

Use a compass to draw a circle centered on the origin of an X-Y coordinate system. Then draw a vector from the origin to a length equal to the radius of the circle. Draw pairs of lines, parallel to the coordinate axes, from the point where the vector tip lies on the circle. Where each line-pair intersects an X-Y axis you can plot a point whose distance from the origin is the rectangular component of the vector. Try this for several different vector orientations around the circle to see how the magnitude and algebraic sign of the rectangular components varies with vector orientation.

Spend about an hour playing with paper, straight edge, and compass and you will acquire an intuitive understanding of vectors that will serve you well for the rest of your studies in math and physics.

Answer 5

no

They are calculated by multiplying the magnitude by the sin or cos of the angle, and sin and cos are never greater than 1.

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