Scalar And Vector Trivia

When multiplying a vector by a scalar (a single numerical value), the result is another vector where each component of the original vector is multiplied by the scalar. For example, if v=(vx​,vy​,vz​) and a is a scalar, then av= (avx,avy,avz).
When dividing a vector by a scalar, each component of the vector is divided by the scalar. This operation results in a scalar quantity.
Therefore, multiplying vectors by scalars results in vectors, and dividing vectors by scalars results in scalars.

The duck is flying south at a speed of 10.0 m/s, but it is flying against a gust of wind with a speed of 2.5 m/s. To find the resultant velocity, subtract the wind's speed from the duck's speed:
 


Resultant velocity=10.0 m/s−2.5 m/s=7.5 m/s south

 

Therefore, the resultant velocity of the duck is 7.5 m/s south. This calculation accounts for the wind's resistance, reducing the duck's effective speed.

 

In a coordinate system, when a vector is oriented at an angle θ with respect to the x-axis, its y-component can be found using the sine function. The y-component (V_y) is equal to the vector's magnitude (V) multiplied by the sine of the angle (θ):
 

Vy​=V sinθ

 


This relationship comes from the definition of the sine function in a right triangle, where the opposite side (y-component) is equal to the hypotenuse (vector's magnitude) multiplied by the sine of the angle. This is a fundamental concept in vector decomposition.

In a coordinate system, when a vector is oriented at an angle θ with respect to the x-axis, its x-component can be found using the cosine function. The x-component (V_x) is equal to the vector's magnitude (V) multiplied by the cosine of the angle (θ):
 

Vx​=V cosθ

 


This relationship comes from the definition of the cosine function in a right triangle, where the adjacent side (x-component) is equal to the hypotenuse (vector's magnitude) multiplied by the cosine of the angle. This is a fundamental concept in vector decomposition.

In the absence of air resistance, a projectile follows a parabolic path. This is due to the influence of gravity acting downward while the projectile moves forward. The horizontal motion is uniform, meaning the horizontal velocity remains constant, while the vertical motion is uniformly accelerated, meaning the vertical velocity changes at a constant rate due to gravity. The combination of these motions results in a curved, parabolic trajectory, described mathematically by a quadratic equation. This predictable path is a key concept in physics, particularly in the study of projectile motion.

From the teacher's perspective, the chalk appears to fall straight down. This is because both the teacher and the chalk are moving forward at the same speed (1.5 m/s). When the chalk is released, it retains this horizontal motion. Since there is no relative horizontal movement between the teacher and the chalk, the only visible motion from the teacher's point of view is the vertical drop due to gravity. To an observer on the ground, the chalk would follow a parabolic path, but to the teacher, it seems to fall straight down.

The distance traveled by the car, 100 kilometers to the north, is classified as a vector quantity. Vector quantities have both magnitude (amount) and direction. In this case:Magnitude: 100 kilometers (the distance traveled)Direction: NorthTherefore, because the travel distance includes a specific direction (north), it falls under the category of vector quantities. Scalars, in contrast, have magnitude only (e.g., speed, time). Hence, the correct answer is Vector.

A force of 10 Newtons applied to push a box to the right is classified as a vector quantity. Vectors have both magnitude and direction. In this case, the magnitude of the force is 10 Newtons, and the direction is to the right. Scalars, in contrast, have only magnitude and no direction (e.g., temperature, mass). Unitless and dimensionless quantities lack units or dimensions and are typically used for pure numbers or ratios. Therefore, the correct answer is B) Vector.

Mass is a scalar quantity. Scalar quantities have magnitude (amount) only and no direction. In the case of mass, whether an object is stationary or in motion, its mass remains the same, and it is measured in kilograms (kg). Vector quantities, on the other hand, have both magnitude and direction (e.g., force, velocity). Mass does not have a direction associated with it, so it is classified as a scalar quantity. Therefore, the correct answer is A) Scalar.