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Questions and Answers
Given the vector function $\mathbf{r}(t) = \langle t^2, 3t, t^3 \rangle$, find the equation of the tangent line to the curve at $t = 2$.
Given the vector function $\mathbf{r}(t) = \langle t^2, 3t, t^3 \rangle$, find the equation of the tangent line to the curve at $t = 2$.
$\mathbf{L}(t) = \langle 4, 6, 8 \rangle + t \langle 4, 3, 12 \rangle$
Determine the domain of the vector function $\mathbf{r}(t) = \langle \sqrt{t-1}, \ln(5-t), \frac{1}{t-3} \rangle$.
Determine the domain of the vector function $\mathbf{r}(t) = \langle \sqrt{t-1}, \ln(5-t), \frac{1}{t-3} \rangle$.
$[1, 3) \cup (3, 5)$
A projectile is fired with an initial velocity of $\mathbf{v}_0 = \langle 10, 20 \rangle$ m/s from an initial position of $\mathbf{r}_0 = \langle 0, 1 \rangle$ meters. Assuming only gravity acts on the projectile ($\mathbf{a}(t) = \langle 0, -9.8 \rangle$ m/s$^2$), find the position vector function $\mathbf{r}(t)$.
A projectile is fired with an initial velocity of $\mathbf{v}_0 = \langle 10, 20 \rangle$ m/s from an initial position of $\mathbf{r}_0 = \langle 0, 1 \rangle$ meters. Assuming only gravity acts on the projectile ($\mathbf{a}(t) = \langle 0, -9.8 \rangle$ m/s$^2$), find the position vector function $\mathbf{r}(t)$.
$\mathbf{r}(t) = \langle 10t, -4.9t^2 + 20t + 1 \rangle$
The position of a particle is given by $\mathbf{r}(t) = \langle \cos(2t), \sin(2t), t \rangle$. Find the speed of the particle at $t = \frac{\pi}{4}$.
The position of a particle is given by $\mathbf{r}(t) = \langle \cos(2t), \sin(2t), t \rangle$. Find the speed of the particle at $t = \frac{\pi}{4}$.
Given the acceleration vector $\mathbf{a}(t) = \langle 0, -10 \rangle$ and initial velocity $\mathbf{v}(0) = \langle 2, 0 \rangle$, find the velocity vector function $\mathbf{v}(t)$.
Given the acceleration vector $\mathbf{a}(t) = \langle 0, -10 \rangle$ and initial velocity $\mathbf{v}(0) = \langle 2, 0 \rangle$, find the velocity vector function $\mathbf{v}(t)$.
Find the limit: $\lim_{t \to 0} \langle \frac{\sin(t)}{t}, e^{-t}, \cos(t) \rangle$.
Find the limit: $\lim_{t \to 0} \langle \frac{\sin(t)}{t}, e^{-t}, \cos(t) \rangle$.
The position of a particle is given by $\mathbf{r}(t) = \langle t^2, t^3 \rangle$. Find the tangential component of acceleration, $a_T$, at $t=1$.
The position of a particle is given by $\mathbf{r}(t) = \langle t^2, t^3 \rangle$. Find the tangential component of acceleration, $a_T$, at $t=1$.
The position of a particle is given by $\mathbf{r}(t) = \langle t, t^2, t^3 \rangle$. Find the unit tangent vector $\mathbf{T}(t)$.
The position of a particle is given by $\mathbf{r}(t) = \langle t, t^2, t^3 \rangle$. Find the unit tangent vector $\mathbf{T}(t)$.
Flashcards
Vector Function
Vector Function
A function that returns a vector, often in the form r(t) = x(t)i + y(t)j + z(t)k.
Domain of a Vector Function
Domain of a Vector Function
The set of all possible input values (t-values) for which the vector function is defined. Found by intersecting the domains of the component functions.
Vector Equation of a Circle
Vector Equation of a Circle
r(t) = Rcos(βt)i + Rsin(βt)j, where R is radius and β affects speed.
Vector Equation of an Ellipse
Vector Equation of an Ellipse
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Limit of a Vector Function
Limit of a Vector Function
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Derivative of a Vector Function
Derivative of a Vector Function
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Antiderivative of a Vector Function
Antiderivative of a Vector Function
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Velocity Vector
Velocity Vector
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Study Notes
- Vector functions can be written as r(t) = x(t)i + y(t)j + z(t)k.
- The domain of a vector function includes all possible t-values and is the intersection of the domains of its component functions.
- The graph of r(t) = x(t)i + y(t)j + z(t)k is defined by the terminal points of all vectors when drawn in standard position, equivalent to graphing the parametric equations x = x(t), y = y(t), z = z(t).
- A circle of radius R centered at the origin has a vector form of r(t) = Rcos(ßt)i + Rsin(ßt)j, with a period of 2π/ß for the circular motion.
Ellipses
- An ellipse x²/A² + y²/D² = 1 can be written as r(t) = Acos(ßt)i + Dsin(ßt)j, having a period of 2π/ß for the elliptical motion.
Vector Function Limits
- If r(t) = x(t)i + y(t)j + z(t)k, then Lim(t→c) r(t) = Lim(t→c) x(t)i + Lim(t→c) y(t)j + Lim(t→c) z(t)k.
Vector Function Derivatives
- If r(t) = x(t)i + y(t)j + z(t)k, the derivative is r'(t) = x'(t)i + y'(t)j + z'(t)k.
Vector Function Integrals
- If r(t) = x(t)i + y(t)j + z(t)k, then ∫r(t) dt = ∫x(t)dti + ∫y(t)dtj + ∫z(t)dt k.
- The vector r'(tâ‚€) is tangent to the graph of r(t) at the point (x(tâ‚€), y(tâ‚€), z(tâ‚€)).
- If r(t) is the position function, then r'(t) = v(t) represents the velocity vector function.
- r"(t) = v'(t) = a(t) represents the acceleration vector function.
- Speed is represented by ||r'(t)|| = ||v(t)||.
- The integral of the acceleration vector function is ∫a(t) dt = v(t) + c.
- The integral of the velocity vector function is ∫v(t) dt = r(t) + c.
Vector Function Rules
- d/dt [f(t)r(t)] = f'(t)r(t) + f(t)r'(t)
- d/dt [G(t) • H(t)] = [G'(t) • H(t)] + [G(t) • H'(t)]
- d/dt [G(t) × H(t)] = [G'(t) × H(t)] + [G(t) × H'(t)]
- Acceleration vector a(t) = a_T(t)T(t) + a_N(t)N(t).
- The unit tangent vector is given by T(t) = r'(t) / ||r'(t)||.
- The unit normal vector is N(t) = T'(t) / ||T'(t)|| = B(t) × T(t).
- The binormal vector is B(t) = T(t) × N(t) = (r'(t) × r"(t)) / ||r'(t) × r"(t)||.
- The tangential component of acceleration is a_T(t) = a(t) • T(t) = d/dt(||v(t)||).
- The normal component of acceleration is a_N(t) = √(||a(t)||² - (a_T(t))²).
- Curvature is defined as κ(t) = ||r'(t) × r"(t)|| / ||r'(t)||³.
- Torsion is defined as τ(t) = [r"'(t) • (r'(t) × r"(t))] / ||r'(t) × r"(t)||².
- Arc length is calculated as ∫ ||v(t)|| dt.
Projectile Motion
- For projectile motion with initial height h₀, initial speed v₀, and angle θ, with downward acceleration due to gravity g:
- a(t) = 0i + (-g)j
- v(t) = (v₀cosθ)i + (-gt + v₀sinθ)j
- r(t) = (v₀cosθ)t i + (-½gt² + (v₀sinθ)t + h₀)j
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Description
Learn about vector functions, including how to write them, determine their domains, and graph them. Explore circles and ellipses in vector form, and understand how to find limits, derivatives, and integrals of vector functions.
