# If a vector is a combination of two unit vectors, is this vector still a unit vector?

If a vector is a combination of two unit vectors, is this vector still a unit vector?

For example $$\vec v_1$$ and $$\vec v_2$$ are both unit vectors, that is, their lengths are both $$1$$.

Now if there is a vector $$\vec f$$.

$$\vec f=\alpha \vec v_1+(1-\alpha )\vec v_2$$ , $$0<\alpha<1$$

Is $$\vec f$$ also a unit vector?

• You've accepted a good answer. I think you could have discovered it yourself if you'd drawn pictures of a few examples in the plane. That's usually a good strategy before you start trying to work with algebra and precise definitions. Oct 15 '19 at 12:43

No. For instance, take:

$$(1,0),(0,1),\alpha=1/2$$

What is $$\vec{f}$$?

Note that $$||\vec{f}||\neq 1$$.

Is it clear? Good studies!

We can check the conditions. So let $$\vec{v}$$ be a vector, being a lineair combination of 2 unit vectors $$\vec{u}_1$$ and $$\vec{u}_2$$. That is:

$$\vec{v} = \alpha \vec{u}_1 + \beta \vec{u}_2$$

Of course, here we suppose we are working in a 2D vectorspace spanned by the vectors $$\vec{u}_1$$ and $$\vec{u}_2$$. Taking the norm gives:

$$||\vec{v}|| = \sqrt{\vec{v} \cdot \vec{v}} = \sqrt{\alpha^2 + \beta^2}$$

The last gives the condition for $$\vec{v}$$ being a unit vector.