# State true or false ( I am not sure what i did wrong)

1. For 𝐮,𝐯 ∈ ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮+𝐯‖.

2. The dot product of two vectors is a vector.

3. For 𝐮,𝐯∈ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮‖+‖𝐯‖.

4. A homogeneous system of linear equations with more equations than variables will always have at least one parameter in its solution.

5. Given a non-zero vector 𝐯, there exist exactly two unit vectors that are parallel to 𝐯.

1. FALSE because if we assumed that a= (-1,-2) and b= (3,4) it would make the statement false
2. FALSE because the dot product of 2 vectors is a scalar
3. FALSE this would have the same assumption as for question 1
4. FALSE I am not sure
5. TRUE I am not sure

I am not sure which one of my answers is/are wrong

• Look at 3, call $-{\bf v} = {\bf w}$ and look up the triangle inequality. – David G. Stork May 19 at 20:53
• wouldn't it be the same for question 1 since the triangle inequality states that ‖𝐮+𝐯‖ ≤ ‖𝐮‖+‖𝐯‖? – Drake May 19 at 21:09

$$1$$ and $$2$$ are both right.

$$3$$ is wrong. The triangle inequality actually implies $$3$$:

$$||u-v||\leq ||u||+||-v||=||u||+||v||$$

$$4$$ is right. Just consider

$$\left\{ \begin{array}{ll} x=0 \\ x=0 \end{array} \right.$$ The only solution is $$x=0$$. This statement would be true the other way around: a homogeneous system of linear equations with more variables than equations will always have at least one parameter in its solution.

And $$5$$ is also right: $$\bf u=\frac{v}{||v||}$$ is a unit vector. Any other vector parallel to $$\bf v$$ (and thus also parallel to $$\bf u$$) is of the form $$k\bf u$$ for some real number $$k$$. And $$k\bf u$$ is a unit vector if and only if $$k=\pm 1$$. Hence the two unit vectors are $$\pm \bf u$$.

• thanks for the explanation – Drake May 19 at 22:32

Number 3 is incorrect. Why? Because of the well-known fact that $$|\bf{x} + \bf{y}| \le |\bf{x}| + |\bf{y}|$$ (the Triangle Inequality).

In particular, $$|\bf{u} - \bf{v}| = |\bf{u} + (- \bf{v})| \le |\bf{u}| + |- \bf{v}| = |\bf{u}| + |\bf{v}|$$.

• thanks for the explanation – Drake May 19 at 22:32