Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am having difficulty solving a problem presented to me by a fellow classmate. I was given a diagram, but I will describe the problem without one, as it is not necessary to fully understand the problem:

Let v and u be an arbitrary vectors in 3 dimensions. If the angle between v and u is $\lambda$ and u makes an angle $\theta$ with the positive x axis ($cos\alpha=\theta$ for u), show that the direction cosines for v can be expressed as

$cos\alpha=cos\lambda cos\theta$

$cos\beta=cos\lambda sin\theta$


where $cos\alpha$, $cos\beta$, and $cos\gamma$ are the angles v makes with the positive x, y, and z axes respectively.

I approached this problem using the formula for the angle between to vectors, since it gives a easily manageable expression for $cos\lambda$, but I keep arriving at fallacious statements. Any help would be appreciated. Thank you.

share|improve this question

1 Answer 1

up vote 3 down vote accepted

The statement is false. Let $\vec{u}=(a,0,0)$ and let $\vec{v}=(0,b,c)$ for any $a,b,c\in\mathbb{R}$, $a>0$. Then $$\cos(\theta)=\frac{\vec{u}\cdot\hat{i}}{||\vec{u}||\cdot||\hat{i}||}=\frac{a\cdot1+0\cdot0+0\cdot0}{|a|\cdot1}=1$$ (pictorially, we can see that $\theta$, the angle between $\vec{u}$ and the positive $x$-axis, is $0$) and $$\cos(\lambda)=\frac{\vec{u}\cdot\vec{v}}{||\vec{u}||\cdot||\vec{v}||}=\frac{a0+0b+0c}{|a|\cdot\sqrt{b^2+c^2}}=0,$$ (pictorially, we can see that $\lambda$, the angle between $\vec{u}$ and $\vec{v}$, is $\frac{\pi}{2}$). According to the statement this would imply that $$\cos(\alpha)=\cos(\lambda)\cos(\theta)=0\cdot1=0$$ $$\cos(\beta)=\cos(\lambda)\sin(\theta)=0\cdot0=0$$ $$\cos(\gamma)=\sin(\lambda)=1$$ so that $\vec{v}$ is necessarily along the positive $z$-axis... which is not necessarily true.

So, a specific counterexample to the statement is $\vec{u}=(1,0,0)$ and $\vec{v}=(0,1,0)$.

Intuitively, we can see that no such statement could be true: let's choose our $\vec{u}\in\mathbb{R}^3$ first. This determines the value of $\theta$, and hence also the value of $\cos(\theta)$ and $\sin(\theta)$. If the formulas $$\cos(\alpha)=\cos(\lambda)\cos(\theta)$$ $$\cos(\beta)=\cos(\lambda)\sin(\theta)$$ $$\cos(\gamma)=\sin(\lambda)$$ were true, then all that would be necessary to determine the direction cosines $\cos(\alpha)$, $\cos(\beta)$, $\cos(\gamma)$ of any $\vec{v}\in\mathbb{R}^3$ would be the angle between $\vec{u}$ and $\vec{v}$ (namely, $\lambda$); but this can't be true, because the set of vectors in $\mathbb{R}^3$ having angle $\lambda$ with $\vec{u}$ forms a cone of vectors, all having different direction cosines in all three directions (so knowing $\lambda$ can't be enough).

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.