Vector Properties and Products Quiz

Properties of Vectors

• Vectors can have a magnitude of zero and are defined in an arbitrary direction.
• Vector addition follows the rule: $\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{c}$.
• Vector subtraction is represented as: $\overrightarrow{a} - \overrightarrow{b} = \overrightarrow{c}$.
• A scalar multiplied by a vector results in the same vector: $a\overrightarrow{a} = \overrightarrow{a}$.
• The dot product of a vector with itself is zero: $\overrightarrow{a} \cdot \overrightarrow{a} = 0$ if $\overrightarrow{a}$ is zero.
• Cross product of a vector with itself is also zero: $\overrightarrow{a} \times \overrightarrow{a} = 0$.

Product of Vectors

• Two primary types of vector products exist:
  • Scalar product (dot product): results in a scalar.
  • Vector product (cross product): results in a vector.

Equal Vectors

• Vectors are equal if they have the same magnitude and direction, denoted as $\overrightarrow{a} = \overrightarrow{b}$.

Negative of a Vector

• A vector can be considered the negative of another if they share the same magnitude but point in opposite directions: $\overrightarrow{a} = -\overrightarrow{b}$.

Modulus of a Vector

• The modulus indicates the magnitude of a vector:
  • For a vector $\overrightarrow{a}$, it is denoted as $|\overrightarrow{a}| = a$.

Unit Vector

• A unit vector has a magnitude of one and is oriented in the same direction as the given vector.
• The unit vector of $\overrightarrow{a}$ is denoted as $\hat{a}$: $|\hat{a}| = 1$.
• The unit vector for another vector $\overrightarrow{b}$ is given by $\hat{b} = \frac{\overrightarrow{b}}{|\overrightarrow{b}|}$.

Orthogonal Unit Vectors

• Orthogonal unit vectors are perpendicular to each other.
• Standard orthogonal unit vectors are:
  • $\hat{i}$: along x-axis
  • $\hat{j}$: along y-axis
  • $\hat{k}$: along z-axis
• Each orthogonal unit vector has a magnitude of 1: $|\hat{i}| = 1$, $|\hat{j}| = 1$, $|\hat{k}| = 1$.

Position Vector

• The position vector of a point 'P' relative to 'O' is represented as:
  • $OP = \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.

Resultant Vector

• The resultant vector is formed when multiple vectors are summed together, denoted as $\vec{R} = \vec{A} + \vec{B}$.
• It is determined by connecting the starting point of the first vector to the endpoint of the last vector.

Displacement Vector

• A displacement vector connects an initial point to a final point, indicating direction toward the final point.
• Position vectors for points 'P' and 'Q' are:
  • $\overrightarrow{OP} = \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$
  • $\overrightarrow{OQ} = \vec{r_1} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$.
• In triangle OPQ, the relationship is: $OQ = OP + PQ$.

Scalars and Vectors

• Physical quantities are measurable and categorized into:
  • Scalar quantities: have only magnitude (e.g., mass, distance).
  • Vector quantities: have both magnitude and direction (e.g., velocity, force).

Representation of a Vector

• Vectors are visually represented as straight lines with arrowheads.
• Examples of vectors include displacement $\overrightarrow{S}$, force $\overrightarrow{F}$, velocity $\overrightarrow{V}$, and acceleration $\overrightarrow{a}$.

Types of Vectors

• Null/Zero Vector: has zero magnitude and an arbitrary direction, represented as $\overrightarrow{0}$.
• Examples include $\overrightarrow{AA} = \overrightarrow{0}$ and $\overrightarrow{BB} = \overrightarrow{0}$.