Scalars and Vectors: Definition, Vector Notation, Differences

Ans. Electric current flows in the opposite direction of electron flow. Current has both magnitude and direction, but it does not follow the vector addition rule. As a result, it's a scalar.

Ans. Even though a force has both magnitude and direction, it is still a force. However, the magnitude of some forces (such as the gravitational force) can be defined as a scalar quantity. To put it another way, while the gravitational force exerted on a particle is not a scalar, its magnitude is.

Ans. There is currently no defined definition of time. Time, however, is not a vector quantity, but rather a scalar one, according to the scientific community. Why? It goes forward in a straight line because its direction never changes.

Ans. It's a hard question, to say the least. I mean, depending on how the temperature is measured, it can be either a scalar or a vector quantity.

It's a scalar quantity, for example, if you're measuring a constant temperature. The measurement of a temperature decline or increase, on the other hand, is a vector quantity.

Ans. Dot product and cross product are distinguished by the fact that the dot product is the product of the magnitude of the vectors and the cos of the angle between them, but the cross product is referred to as the product of the magnitude of the vector and the sine of the angle in which they subtend each other.

Ans. A unit vector has a magnitude of one. There are no units or dimensions in a unit vector.

Ans. When two vectors are headed in the same direction, the maximum sum of the two vectors is obtained. When the two vectors are headed in opposing directions, the smallest sum is obtained.

Ans. Vector addition does not work with any two vectors. Only two vectors of the same type and nature are combined. Two velocity vectors, for example, can be combined, but one velocity vector and one force vector cannot.

Ans. The associative property of vector addition asserts that regardless of the order in which the vectors are placed, the sum of the vectors remains the same.

Ans. Yes, the sum of two vectors of identical magnitude pointing in opposing directions is zero.

Ans. The key difference between scalar and vector quantity is direction. Scalar Quantity has magnitude only and no direction. A vector quantity, on the other hand, has a magnitude and direction.

Ans. Both distance and displacement are measurements of length. Distance is the total length covered by a body, whereas displacement is the shortest distance between two endpoints. Where distance is just the length, displacement is along a certain direction, giving the shortest length between the starting and the endpoint. Thus, distance only has a value, but displacement has a direction along with a value.

Ans. a) Parallelogram law of vector addition

This law states that if two vectors acting at a point can be represented in magnitude and direction by the two adjacent sides of a parallelogram, then the diagonal of the parallelogram through that point represents the resultant vector.

b) R = \(\sqrt{A^2+B^2+2ABcos \theta}\)

Here R = F, A = F and B = F

F = \(\sqrt{F^2+F^2+2F^2 cos\theta}\)

F² = F²+ F²+ 2F²Cos θ

F² = 2F² + 2F² Cos θ

F² – 2F² = 2F² Cos θ

F² = 2F² Cos θ

-1/2 = Cos θ

θ = 120°

Ans. a) \(\hat{i}, \hat{j}, \hat{k}\) are orthogonal unit vectors.

b) A vector having zero magnitudes is called a zero vector.

Since the magnitude is zero, we don’t have to specify its direction.

Example: Suppose that an object which is at p at time t moves to p’ and then comes back to p. In this case, displacement is a null vector.

Ans. a) Parallelogram

b) Parallelogram law of vector addition: This law states that if two vectors acting at a point can be represented in magnitude and direction by the two adjacent sides of a parallelogram, then the diagonal of the parallelogram through that point represents the resultant vector.

c) Vertical height of body is decided by vertical com-ponent of velocity (u sinθ). The vertical displace-ment of projectile can be found using the formula

v² = u² + 2as

When we substitute v = 0, a = – g,s = H and u = usinθ, we get

0 = (usinθ)² + 2 – g x H

2gH = u²sin²θ

H = \(\frac{u²sin²θ}{2g}\)

Ans. The difference between a scalar and a vector is that a vector requires a direction. Scalar quantities have only magnitude; vector quantities have both magnitude and direction. Time is completely separated from direction; it is a scalar. It has only magnitude, no direction.

Important differential factors:

• Speed is a scalar, while velocity is a vector.
• Distance is a scalar, while displacement is a vector.
• Force and acceleration are vectors. Time is a scalar.

Ans. Displacement is a vector quantity; the direction that Michael travels will be either positive or negative along an axis. We are being asked to solve for his position relative to his starting point, NOT for the distance he has walked.

First we need to find his total distance travelled along the y-axis. Let's say that all of his movement north is positive and south is negative.

∑Y=10m−5m=5m. He moved a net of 5 meters to the north along the y-axis.

Now let's do the same for the x-axis, using positive for east and negative for west.

∑X=−3m+12m=9m. He moved a net of 9 meters to the east.

Now to find the resultant displacement, we use the Pythagorean Theorem. The net movement north will be perpendicular to the net movement east, forming a right triangle. Michael's position relative to his starting point will be the hypotenuse of this triangle.

D² = (∑X)²+(∑Y)²

D² = (9m)²+ (5m)²

D² = 106m²

Now, takeing square root of both sides.

\(\sqrt{D^2} = \sqrt{106m^2}\)

D=10.30m

Since the problem only asks for the magnitude of the displacement, we do not need to provide the direction.

Ans. Distance is a scalar quantity and will take into account only the number of floors travelled, regardless of the direction of movement.

Walter takes an incredibly complicated path to wash the windows on the building. When calculating distance, we add up all the movement he does, regardless of direction.

First, he travels down one floor (4th to 3rd).

D_total=(1)

Then he travels up three floors (3rd to 6th).

D_total=(1)+(3)

Then he travels down one floor (6th to 5th), then down another three floors (5th to 2nd).

D_total=(1)+(3)+(1)+(3)

Finally, he travels down one more floor (2nd to 1st).

D_total=(1)+(3)+(1)+(3)+(1)=9

In total, Walter travelled 9 floors.