Vectors

Vectors - Direction and Magnitude

UNIT Vectors

Vector Quantities

A vector is a physical quantity that has both magnitude and direction. For example, if someone says an object is moving at 10 m/s, that only tells us how fast it is moving. However, if they say the object is moving 10 m/s to the right, we now know both the magnitude and the direction, making it a vector quantity.

Vectors are extremely important in physics because many physical quantities depend on direction. Examples of vector quantities include:

• Displacement
• Velocity
• Acceleration
• Force
• Momentum

These are different than scalar quanttites because their direction affects how they interact and combine.

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Scalar Quantaties

A scalar quantity only has a magnitude. Examples of scalr quantities include:

• Time
• Speed
• Mass
• Energy
• Temperature

For example, saying an object has a mass of 5 kg or a temperature of 20°C fully describes the quantity without needing a direction. Vectors, however, must always include direction. Saying a force is 20 N is incomplete. Instead, it must be described as something like 20 N upward or 20 N to the left.

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Representing Vectors

Vectors are typically represented by an arrow. These arrows have two import parts:

• length of the arrow → represents the magnitude of the vector
• direction of the arrow → represents the direction of the vector

Longer arrows represents larger magnitudes, while shorter arrows represent smaller magnitudes. The direction just means the direction of the vector

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Components of Vectors

Someimte vectors are at an angle, so it is helpfull to break them down into the vertical and horizontal components.
The components tell how much a vectors points in each direction.

For example, a vector V at an angle θ can be broken into two parts. V_x and V_y:

V_x = V cos(θ)

V_y = V sin(θ)

Where:

• V is the magnitude of the vector
• θ is the angle the vector makes with the horizontal axis
• V_x is the horizontal component of the vector
• V_y is the vertical component of the vector

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Vector Addition

In physics, vectors must be added differently than regular numbers because direction matters. If two vectors point in the same direction, their magnitudes add normally. However, if they point in different directions, their directions must be taken into account when determining the final result.

\( \vec{R} = \vec{A} + \vec{B} \)

Where \( \vec{R} \) represents the resultant vector.

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If two vectors are pointed in the same direction, there magnitudes can simply be added.

Examples:

• \(\vec{A}\) = 5 N to the right
• \(\vec{B}\) = 3 N to the right

\( \vec{R} = 5N +3N = 8N \)

Result: 8 N to the right

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If two vectors are pointed in the opposite direction, there magnitudes subtract

Examples:

• \(\vec{A}\) = 10 N to the right
• \(\vec{B}\) = 3 N to the left

\( \vec{R} = 10N - 3N = 7N \)

Result: 7 N to the right( the direction of the larger vectore )