Question

question #7
reference q.46813
can we add 3 parallel vectors together and get the zero vector (a vector with no magnitude)?

question #8
reference q.46816
can we add 3 non-parallel vectors and get the zero vector (a vector with no magnitude)?

Explanation:

Question #7

Step1: Recall parallel vectors property

Parallel vectors have the same or opposite direction, so they can be represented as scalar multiples of a single vector, say $\vec{v}$. Let the three parallel vectors be $a\vec{v}$, $b\vec{v}$, $c\vec{v}$, where $a, b, c$ are scalars.

Step2: Sum the vectors

Their sum is $(a + b + c)\vec{v}$. For this to be the zero vector, we need $(a + b + c)\vec{v}=\vec{0}$. Since $\vec{v}
eq\vec{0}$ (as it has magnitude), we need $a + b + c = 0$. For example, take $\vec{v}$, $-\frac{1}{2}\vec{v}$, $-\frac{1}{2}\vec{v}$. Their sum is $\vec{v}-\frac{1}{2}\vec{v}-\frac{1}{2}\vec{v}=\vec{0}$. So it is possible.

Step1: Recall vector addition in plane/space

In a plane (2D) or space (3D), non - parallel vectors can form a closed triangle (in 2D) or a closed polygon (in 3D) when added. For three non - parallel vectors $\vec{A}$, $\vec{B}$, $\vec{C}$, we can use the triangle law of vector addition. First, add $\vec{A}$ and $\vec{B}$ to get $\vec{R}=\vec{A}+\vec{B}$. Then, if we can find a vector $\vec{C}=-\vec{R}$, the sum $\vec{A}+\vec{B}+\vec{C}=\vec{0}$. Geometrically, if we have three vectors that can be arranged head - to - tail to form a triangle (in 2D) or a closed figure (in 3D), their sum is zero. For example, in a triangle with sides represented by vectors $\vec{AB}$, $\vec{BC}$, $\vec{CA}$, $\vec{AB}+\vec{BC}+\vec{CA}=\vec{0}$.

Step2: Conclusion

So, we can add three non - parallel vectors to get the zero vector.

Answer:

Yes

Question #8