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4.9 Vector-Valued Functions

1 min readβ€’june 18, 2024

Jesse

Jesse

Jesse

Jesse

4.9 Vector-Valued Functions

Hear me out: Our knowledge of parametric functions, planar motion, and vectors can be combined to represent planar motion in terms of vector-valued functions! πŸ’°

➑️ Position Vector

The position of a particle moving in a two-dimensional plane can be represented by a vector-valued function, p(t) = x(t)i + y(t)j, where x(t) and y(t) are the coordinates of the particle at time t and i and j are the unit vectors in the x- and y-directions, respectively. 🧍

Alternatively, the position vector can also be represented in the form of p(t) = < x(t), y(t) >, where x(t) and y(t) are the components of the vector.

Vectors_ij.png

Source: Numeracy

↗️ Velocity Vector

A vector-valued function, v(t) = < x(t), y(t) >, can be used to express the velocity of a particle moving in a two-dimensional plane at different times, t. The x and y components of the vector represent the horizontal and vertical velocities of the particle, respectively. πŸš„

At any given time, t, the sign of x(t) indicates the direction of the horizontal velocity; a positive value indicates the particle is moving to the right ▢️ and a negative value indicates the particle is moving to the left. ◀️ Similarly, the sign of y(t) indicates the direction of the vertical velocity; a positive value indicates the particle is moving upwards πŸ”Ό and a negative value indicates the particle is moving downwards. πŸ”½

TrigVectorExample1Graph2.png

Source: Xaktly

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πŸ“š

Β >Β 

πŸ“ˆΒ 

Β >Β 

πŸ“

4.9 Vector-Valued Functions

1 min readβ€’june 18, 2024

Jesse

Jesse

Jesse

Jesse

4.9 Vector-Valued Functions

Hear me out: Our knowledge of parametric functions, planar motion, and vectors can be combined to represent planar motion in terms of vector-valued functions! πŸ’°

➑️ Position Vector

The position of a particle moving in a two-dimensional plane can be represented by a vector-valued function, p(t) = x(t)i + y(t)j, where x(t) and y(t) are the coordinates of the particle at time t and i and j are the unit vectors in the x- and y-directions, respectively. 🧍

Alternatively, the position vector can also be represented in the form of p(t) = < x(t), y(t) >, where x(t) and y(t) are the components of the vector.

Vectors_ij.png

Source: Numeracy

↗️ Velocity Vector

A vector-valued function, v(t) = < x(t), y(t) >, can be used to express the velocity of a particle moving in a two-dimensional plane at different times, t. The x and y components of the vector represent the horizontal and vertical velocities of the particle, respectively. πŸš„

At any given time, t, the sign of x(t) indicates the direction of the horizontal velocity; a positive value indicates the particle is moving to the right ▢️ and a negative value indicates the particle is moving to the left. ◀️ Similarly, the sign of y(t) indicates the direction of the vertical velocity; a positive value indicates the particle is moving upwards πŸ”Ό and a negative value indicates the particle is moving downwards. πŸ”½

TrigVectorExample1Graph2.png

Source: Xaktly