Another simple vector question

[-EquinoX-]

Homework Statement

Find a unit vector from the point P = (1, 2) and toward the point Q = (4, 6).

Homework Equations

The Attempt at a Solution

The answer at the back of the book says 3/5i + 4/5j, shouldn't it just be 3i + 4j?

 

[Unco]

The answer at the back of the book says 3/5i + 4/5j, shouldn't it just be 3i + 4j?

A unit vector has length 1.

 

[-EquinoX-]

so what do you mean? If I am asked to find another vector that has the same direction with length 10, how do I do that? if I changed the question to (3,6) instead of (4,6) what would be the answer then

 

[Unco]

so what do you mean? ... if I changed the question to (3,6) instead of (4,6) what would be the answer then

Multiplying a vector by a nonzero scalar does not change its direction. If the length of a nonzero vector v is a, then (1/a)v has length 1.

The following is then an exercise for you.

If I am asked to find another vector that has the same direction with length 10, how do I do that?

 

[-EquinoX-]

ok so if that's so then finding another vector that has the same direction as 3/5i + 4/5j, will be something like

1/10 (3/5i + 4/5j) am I right?

the answer at the back of the book is 6i + 8j, I don't where that came from...

 

[Unco]

ok so if that's so then finding another vector that has the same direction as 3/5i + 4/5j, will be something like

1/10 (3/5i + 4/5j) am I right?

Come on, Equi, what is the length of that vector?

You have $$\textstyle\mathbf{v} = (\frac{3}{5}, \frac{4}{5})$$, which has length

$$.\; \; \; \; \; \textstyle ||\mathbf{v}|| = \left\|\frac{1}{5}(3, 4)\right\| = \frac{1}{5}||(3, 4)|| = \frac{1}{5}\sqrt{3^2+4^2} = \frac{1}{5}5 = 1$$.

Hence $$\textstyle 10\mathbf{v} = 10(\frac{3}{5}, \frac{4}{5}) = (6,8)$$ has length 10 since

$$.\; \; \; \; \; ||10\mathbf{v}|| = 10||\mathbf{v}|| = 10$$.