Geometry
# Properties of a Vector

Which of the following is a vector quantity?

- The displacement from New York to London
- The color of a shoe
- The number of books on a shelf
- An array with the number of books reads by each student in a class
- The speed of an airplane

Which of the following vectors in the regular hexagon below has the same magnitude as \(\vec{AB}\) but opposite direction?

\[\]
**Details and assumptions:**

- O is the intersection point of the three diagonal lines.

If \(\vec{a}=\vec{b},\) which of the following is/are true?

I. \(\vec{a}\) and \(\vec{b}\) are parallel.

II. \(\vec{a}\) and \(\vec{b}\) are equal in magnitude.

III. The initial points of \(\vec{a}\) and \(\vec{b}\) are the same.

IV. The terminal points of \(\vec{a}\) and \(\vec{b}\) are the same.

Let \(\vec{OA}=\vec{a}+2\vec{b}, \vec{OB}=3\vec{a}-\vec{b},\) and \(\vec{OC}=2\vec{a}+k\vec{b}\) in the diagram below. If the three points \(A, B,\) and \(C\) lie on the line \(l\), which of the following is equal to \(k\)?

\[\]
**Details and assumptions:**

- Point \(O\) does \(\color{red}{\text{not}}\) lie on the line \(l.\)

Find the unit vector in the direction \((1,2,3).\)

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