Vectors are physical quantities that require both magnitude and direction for their complete description, such as velocity, force, and acceleration. Represented graphically by directed line segments, the length corresponds to the vector's magnitude, and the arrow indicates its direction. In a Cartesian coordinate system, vectors are resolved into rectangular components along mutually perpendicular axes (x, y, and z). The addition of vectors follows the head-to-tail rule, while subtraction involves reversing the direction of a vector. Multiplying vectors by scalars changes their magnitude but can reverse their direction if the scalar is negative. Other concepts include unit vectors, null vectors, and equal vectors, all of which play fundamental roles in defining directions, representing zero magnitude, and comparing vector properties.
What is a vector?
A vector is a physical quantity that has both magnitude and direction, such as velocity, acceleration, and force.
How is a vector graphically represented?
A vector is represented by a directed line segment, where the length corresponds to its magnitude and the arrow indicates its direction.
What are rectangular components of a vector?
Rectangular components are the projections of a vector along mutually perpendicular axes, such as x, y, and z, in a Cartesian coordinate system.
What is the head-to-tail rule for vector addition?
The head-to-tail rule involves placing the tail of one vector at the head of another. The resultant vector is the line joining the tail of the first vector to the head of the second.
What happens when a vector is multiplied by a scalar?
When a vector is multiplied by a scalar, its magnitude changes by the scalar's value. If the scalar is negative, the direction of the vector is reversed.
Which of the following is an example of a vector quantity?
a) Speed
b) Distance
c) Velocity
d) Temperature
Answer: c) Velocity
What is the magnitude of a vector with components 3 units along the x-axis and 4 units along the y-axis?
a) 5 units
b) 7 units
c) 12 units
d) 9 units
Answer: a) 5 units
Which operation involves adding vectors by aligning the tail of one vector to the head of another?
a) Scalar multiplication
b) Head-to-tail rule
c) Cross product
d) Dot product
Answer: b) Head-to-tail rule
A vector that has a magnitude of 1 is called a:
a) Null vector
b) Position vector
c) Unit vector
d) Scalar vector
Answer: c) Unit vector
What happens when a vector is multiplied by a negative scalar?
a) Its magnitude increases.
b) Its direction reverses.
c) It becomes a unit vector.
d) Its magnitude decreases but direction remains the same.
Answer: b) Its direction reverses.