Name: Resolving Vectors Into Components
Uploaded: 2019-07-19
Duration: 5 min 37 s
Description: A vector can be resolved into a number of components according to convenience. Watch this video to learn how this can make our lives easier while solving pro...

Question

A large number of vectors are found to form a closed polygon when placed one after another by parallel displacements. What does that imply?

Accepted Answer

If a set of vectors, taken in a given order, gives a closed polygon, the resultant of all these vectors would be a null vector or zero vector.

The head of any vector meets the tail of the following vector.

Let us call the starting points the vectors $\vec{a_{1}}, \vec{a_{2}}, \vec{a_{3}} . . . . . . . . . . . \vec{a_{n}}$

The order in which they are kept is $\vec{a_{1}}$ , followed by $\vec{a_{2}}$ , followed by $\vec{a_{3}}$ , followed by ⋯⋯followed by $\vec{a_{n}}$ , followed by $\vec{a_{1}}$ .

The head of any vector meets the tail of the following vector.

Let us call the starting points the vectors $\vec{a_{1}}, \vec{a_{2}}, \vec{a_{3}} . . . . . . . . . . . \vec{a_{n}} a s p_{1}, p_{2}, p_{3}, . . . . . . . p_{n}$

Recall the triangle law and the polygon law of vector addition.

The resultant of $\vec{a_{1}}$ and $\vec{a_{2}}$ is a vector from $p_{1} t o p_{3}$ .

The resultant of $\vec{a_{1}}, \vec{a_{2}}$ and $\vec{a_{3}}$ is a vector from $p_{1} t o p_{4}$

The resultant of $\vec{a_{1}}, \vec{a_{2}}, \vec{a_{3}}$ and $\vec{a_{4}}$ is a vector from $p_{1} t o p_{5}$

Similarly, the resultant of $\vec{a_{1}}, \vec{a_{2}}, \vec{a_{3}}, . . . . . . . . . . ., {\vec{a}}_{n - 1}$ and $\vec{a_{n}}$ is a vector from $p_{1} t o p_{n}$ . Since the starting and ending positions of this vector is the same, it has zero magnitude and is therefore a null vector.

A large number of vectors are found to form a closed polygon when placed one after another by parallel displacements. What does that imply?

Important Questions on Scalars and Vectors

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