# Write F as a sum of an orthogonal and parallel vector

an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion.

the parallel would be $$\hat{i}+\hat{j}$$ and the orthogonal would be $$\hat{i} - \hat{j}$$

using projection of F onto the parallel and orthogonal

$$\frac{<1,1>\cdot<2,1>}{||<1,1>||^2}<1,1> = <\frac{3}{2}, \frac{3}{2} >$$

$$\frac{<1,-1>\cdot<2,1>}{||<1,-1>||^2}<1,-1> = <\frac{1}{2} , \frac{-1}{2}>$$

$$\vec{F} = <\frac{3}{2}, \frac{3}{2} > + <\frac{1}{2}, \frac{-1}{2} >$$

$$= 2\hat{i} + 1\hat{j}$$

Related Introductory Physics Homework Help News on Phys.org
STEMucator
Homework Helper
an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion.
Out of curiosity, what is the i+j direction? Do you mean the positive x and y directions? Where is the object exactly?

I'm assuming a few things here, but:

Place the force vector ##\vec F = 2 \hat i + 1 \hat j## at the position of the object; Then find a unit vector ##\vec u## in the direction of the object (using a position vector ##\vec r##).

Use ##\vec F## and ##\vec u## to find the parallel projection ##F_{||}## (by using the dot product). Take a moment to express the parallel component in vector form, ##\vec F_{||}##, by using ##\vec u##.

The perpendicular component is easily obtained by geometry afterwards, i.e:

##\vec F_{\perp} = \vec F - \vec F_{||}##

ehild
Homework Helper
an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion.

the parallel would be $$\hat{i}+\hat{j}$$ and the orthogonal would be $$\hat{i} - \hat{j}$$
I know what you mean, but the sentence is not correct. It means as if the parallel and perpendicular components of the force vector would be i+j and i-j. You have to find the force component parallel with the direction of motion , say v and perpendicular to it. The direction of the motion is parallel to i+j and i-j is perpendicular to it.

using projection of F onto the parallel and orthogonal /onto the direction of v and onto the direction orthogonal to v

$$\frac{<1,1>\cdot<2,1>}{||<1,1>||^2}<1,1> = <\frac{3}{2}, \frac{3}{2} >$$

$$\frac{<1,-1>\cdot<2,1>}{||<1,-1>||^2}<1,-1> = <\frac{1}{2} , \frac{-1}{2}>$$

$$\vec{F} = <\frac{3}{2}, \frac{3}{2} > + <\frac{1}{2}, \frac{-1}{2} >$$

$$= 2\hat{i} + 1\hat{j}$$
The result is correct.

ehild

Out of curiosity, what is the i+j direction? Do you mean the positive x and y directions? Where is the object exactly?

I'm assuming a few things here, but:

Place the force vector ##\vec F = 2 \hat i + 1 \hat j## at the position of the object; Then find a unit vector ##\vec u## in the direction of the object (using a position vector ##\vec r##).

Use ##\vec F## and ##\vec u## to find the parallel projection ##F_{||}## (by using the dot product). Take a moment to express the parallel component in vector form, ##\vec F_{||}##, by using ##\vec u##.

The perpendicular component is easily obtained by geometry afterwards, i.e:

##\vec F_{\perp} = \vec F - \vec F_{||}##
thats the only information given. this is in a calc class not a physics class so like most math classes they give the bare minimum with physics problems

I know what you mean, but the sentence is not correct. It means as if the parallel and perpendicular components of the force vector would be i+j and i-j. You have to find the force component parallel with the direction of motion , say v and perpendicular to it. The direction of the motion is parallel to i+j and i-j is perpendicular to it.

The result is correct.

ehild

yeah I typed the question word for word, not the best wording

ehild
Homework Helper
yeah I typed the question word for word, not the best wording
The question was correct. It was the first sentence of your solution which was not.

ehild