Vector and Matrix Quantities: Understanding Vector Quantities and Representation (CCSS.N-VM.1)
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Common Core High School Number And Quantity › Vector and Matrix Quantities: Understanding Vector Quantities and Representation (CCSS.N-VM.1)
In physics, quantities like displacement, velocity, and force are used to describe motion.
Which statement defines a vector?
A quantity with magnitude only.
A quantity with direction only.
A quantity with both magnitude and direction.
A label used to name an object.
Explanation
A vector has both magnitude and direction; a scalar has only magnitude. Symbols like $\vec{v}$ denote a vector, and $|\vec{v}|$ or $|\vec{v}|$ denote its magnitude.
A car moves in a straight line at a constant speed of 20 m/s toward the west.
Which statement represents its velocity vector?
20 m/s.
20 m/s west.
West.
20 m.
Explanation
Velocity is a vector: it includes magnitude and direction. Option B provides both; A lacks direction; C lacks magnitude; D is a distance, not a velocity.
Two horizontal forces act on a box at the same time: 10 N to the right and 4 N to the left.
What is the resultant force vector on the box?
6 N to the right.
14 N.
6 N to the left.
10 N to the right and 4 N to the left.
Explanation
Forces add as vectors. The leftward 4 N opposes the rightward 10 N, giving a net of $10-4=6$ N to the right. Option B ignores direction; C has the wrong direction; D lists components, not the resultant.
A lifeguard reports that the ocean current moves swimmers at 1.5 m/s to the east. Which statement defines a vector?
A quantity with magnitude only.
A number that names a position.
A quantity with both magnitude and direction.
A description of a path traveled.
Explanation
A vector has both magnitude and direction, often written as $v$; its magnitude is $|v|$ or $|v|$. Scalars, like speed without direction, have magnitude only.
A cyclist maintains a steady motion along a straight road heading due south. Which is a scalar, not a vector?
Speed
Velocity
Displacement
Force
Explanation
Speed is scalar (magnitude only). Velocity, displacement, and force require both magnitude and direction, fitting the vector idea $v$ with magnitude $|v|$ or $|v|$.
A hiker leaves camp and ends up 3 km east and 4 km north of the starting point. Which vector best represents this displacement?
7 km northeast
5 km due east
Move 3 km east, then 4 km north (two separate moves)
About 5 km at 53° north of east
Explanation
The resultant displacement is a single vector with magnitude 5 km (by the Pythagorean theorem) and a direction about 53° north of east. Options A and B ignore correct direction or magnitude; C describes a path, not one vector.
You push a box with a force of 10 N to the right. Which is a scalar, not a vector?
Force on the box
Mass of the box
Acceleration of the box
Displacement of the box
Explanation
Mass is scalar (magnitude only). Force, acceleration (units $\text{m/s}^2$), and displacement are vectors, each needing both magnitude and direction (e.g., $v$ with magnitude $|v|$ or $|v|$).
After parking, a car is 60 meters west of where it started. Which vector best represents this displacement?
West
60 m/s west
60 meters west
60 meters
Explanation
A displacement vector must include magnitude and direction: 60 meters west. Options A and D are missing either magnitude or direction; B is a velocity (wrong quantity).