# What does it mean to find the component of 2 vectors in the direction of another vector?

I understand how to take the cross-product of 2 vectors ($\vec{a}\times\vec{b}$), but what does it mean to find the component of $\vec{a}\times\vec{b}$ along the direction of another vector (for example $\vec{c}$)?

If you're doing the cross product of 2 vectors, are not the resultant components the third vector since the right-hand thumb rule gives a clear picture of the direction of the resultant vector which is perpendicular to both the vectors?

• Would Mathematics be a better home for this question? – Qmechanic Aug 24 '18 at 7:14
• suppose one takes a dot product of the two vectors separately with the given vector then I think he gets components in the direction of the third one.. now it is up to him ... how he can use those components...as an example two forces are acting on a body and it is constrained to move in a fixed direction...then the components of forces in the direction of motion can give a measure of work done by those two forces... – drvrm Aug 24 '18 at 7:30
• @Qmechanic I am unsure. I've seen other questions regarding vectors on the Physics section here. We are going over vectors for electromagnetic fields. – Nava Moore Aug 24 '18 at 7:34
• @drvrm if I take the dot product separately, when do I use the cross product in this scenario? A question asks to take the cross product of two vectors in the direction of another. I assume axb are perpendicular and moving in a 3rd direction like an EM wave. – Nava Moore Aug 24 '18 at 7:40
• @Nava Moore- imagine that a body is falling on earth and earth is spinning then the cross product of angular velocity w vector and the instantaneous velocity of the body defines a force- a vector perpendicular to the plane containing w and v- now this force may have components in the dirction of X and Y axes fixed on the earth- and its physically meaningful as a deflection of falling body can be observed...the force is known as coriolis force. – drvrm Aug 24 '18 at 7:54

## 2 Answers

Suppose $\vec{g}$ to be the vector product $\vec{a}\times\vec{b}$, the meaning of finding the component of the product $\vec{a}\times\vec{b}$ on the direction of another vector - I represented the direction with the versor $\vec{u}$ of the very $\vec{c}$ - is to project that vector $\vec{g}$ on the direction of the vector $\vec{c}$ as you asked, which can be done with the scalar product $\vec{g}\cdot\vec{u}$. • What I don't see in your answer is that at the end the real number $\; (\mathbf{a}\boldsymbol{\times}\mathbf{b})\boldsymbol{\cdot}\mathbf{c}\;$ is by magnitude the volume of the oblique in general parallelepiped formed by $\; \mathbf{a},\mathbf{b},\mathbf{c}\;$ and sign depending on the orientiation of this triad of vectors. – Frobenius Aug 24 '18 at 12:16

With $\vec{d}=\vec{a}\times\vec{b}$, $\vec{d}$ has a component $\vec{d} \cdot \frac{\vec{c}}{|c|}$ in the direction of $\vec{c}$. You can think of $\frac{\vec{c}}{|c|}$ as a unit vector in the direction of $\vec{c}$.

Your second paragraph is correct.

• What I don't see in your answer is that at the end the real number $\; (\mathbf{a}\boldsymbol{\times}\mathbf{b})\boldsymbol{\cdot}\mathbf{c}\;$ is by magnitude the volume of the oblique in general parallelepiped formed by $\; \mathbf{a},\mathbf{b},\mathbf{c}\;$ and sign depending on the orientiation of this triad of vectors. – Frobenius Aug 24 '18 at 12:15