Vector Operations and Their Applications

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Questions and Answers

What is the correct formula for the magnitude of a vector v?

||v|| = v₁ + v₂ + v₃

||v|| = v₁² + v₂² + v₃²

||v|| = √(v₁² + v₂²)

||v|| = √(v₁² + v₂² + v₃²) (correct)

How is the angle between two vectors v and u calculated using their dot product?

cos θ = (u ⋅ v) / (||u|| ||v||) (correct)

sin θ = (v ⋅ u) / (||v|| ||u||)

tan θ = (v ⋅ u) / (||v|| ||u||)

cos θ = (v ⋅ u) / (||v|| + ||u||)

Which equation correctly represents the geometric shape of a sphere?

(x-h)² + (y-k)² + (z-l)² = -r²

(x-h)² + (y-k)² + (z-l)² = r² (correct)

(x-h)² + (y-k)² + (z-l)² = r

(x-h)² + (y-k)² + (z-l)² = √r

What describes two vectors as orthogonal?

Their dot product equals 0. Signup and view all the answers

Which of the following describes the projection of vector a onto vector b?

projba = (a ⋅ b) / ||b||² * b Signup and view all the answers

What is the correct symmetric equation for a line defined by a point (x₁, y₁, z₁) and a direction vector (a, b, c)?

x - x₁ / a = y - y₁ / b = z - z₁ / c Signup and view all the answers

Which equation represents a hyperboloid of one sheet?

x² / a² + y² / b² - z² / c² = 1 Signup and view all the answers

When are two planes considered orthogonal?

Their normal vectors are orthogonal. Signup and view all the answers

What is the formula for finding the tangent vector at a specific point $t_0$ for a vector-valued function?

$oldsymbol{r}'(t_0) = oldsymbol{r}(t_0) + t oldsymbol{r}'(t_0)$ Signup and view all the answers

How is the speed of a particle represented in vector-valued functions?

$||oldsymbol{v}(t)||$ Signup and view all the answers

What does the curvature of a curve represent mathematically?

$K = rac{||oldsymbol{T}'(t)||}{||oldsymbol{r}'(t)||}$ Signup and view all the answers

What is the correct way to evaluate definite integrals of vector-valued functions?

$igint_{a}^{b} oldsymbol{r}(t) ext{d}t = igg< igint_{a}^{b} f(t) ext{d}t, igint_{a}^{b} g(t) ext{d}t, igint_{a}^{b} h(t) ext{d}t igg>$ Signup and view all the answers

What is Newton's second law of motion expressed in vector form?

$oldsymbol{F} = m oldsymbol{a}$ Signup and view all the answers

Which condition indicates that a function is continuous at point (a, b)?

The limit as (x, y) approaches (a, b) equals f(a, b) Signup and view all the answers

What two conditions must be satisfied to classify a critical point as a local minimum?

D > 0 and fxx > 0 Signup and view all the answers

How is the directional derivative of a function f(x, y) represented mathematically?

Duf = ∇f ⋅ u Signup and view all the answers

Which of the following represents the correct form of the chain rule when differentiating z with respect to t?

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) Signup and view all the answers

What does the Two-Path Test assess regarding limits?

If the limit exists for all paths taken to a point Signup and view all the answers

What is the main purpose of Lagrange multipliers in optimization problems?

To find maximum/minimum values of a function subject to a constraint Signup and view all the answers

Which equation represents the double integral over a rectangular region?

$\iint_R f(x, y) , dA = \int_a^b \int_c^d f(x, y) , dy , dx$ Signup and view all the answers

In the context of double integrals, how is the average value of a function determined?

$\frac{1}{A} \iint_R f(x, y) , dA$ where A is the total area of R Signup and view all the answers

What is the correct expression to evaluate a double integral over a general region defined by $g_1(x)$ and $g_2(x)$?

$\iint_R f(x, y) , dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x, y) , dy , dx$ Signup and view all the answers

For the volume integral in cylindrical coordinates, what are the correct variables and differential area element?

Variables: $r, \theta, z$; $dV = r , dr , d\theta , dz$ Signup and view all the answers

When using spherical coordinates, which of the following is the correct representation for the volume differential?

$dV = \rho^2 , d\rho , d\phi , d\theta$ Signup and view all the answers

What condition is typically applied when using Lagrange multipliers to find the extrema of a function?

$\nabla f = \lambda \nabla g$ along with $g(x, y, z) = 0$ must be satisfied Signup and view all the answers

Which of the following describes how to convert from rectangular to polar coordinates in double integrals?

Replace $x$ and $y$ with $r \cos(\theta)$ and $r \sin(\theta)$ and use $dA = r , dr , d\theta$ Signup and view all the answers

Which equation correctly represents the mass of a solid with variable density?

$m = ho(x,y,z) imes dV$ Signup and view all the answers

What is a characteristic of conservative vector fields?

They result in a zero line integral for any closed path. Signup and view all the answers

Which of the following expressions represents Green's theorem?

$egin{equation} ext{Area}{R} = igintigint{R} igg( rac{ ext{d}Q}{ ext{d}x} - rac{ ext{d}P}{ ext{d}y} igg) dA$ Signup and view all the answers

Which statement is true about flow lines in vector fields?

Flow lines are perpendicular to the gradient of the corresponding potential function. Signup and view all the answers

What is the correct representation of a line integral of a vector field?

$ ext{Line integral} = igint_C F ullet dr = igint_C F ullet T imes ds$ Signup and view all the answers

What does the divergence of a vector field measure?

The 'outflowing-ness' at a specific point Signup and view all the answers

Which condition is not required for Stokes' Theorem to apply?

The surface must be a closed surface Signup and view all the answers

How is the flux of a vector field through a closed surface calculated according to the Divergence Theorem?

By integrating the divergence of the field over the volume enclosed by the surface Signup and view all the answers

What does the curl of a vector field indicate?

The twisting or rotation of the vector field Signup and view all the answers

In the context of surface integrals, what does the notation $ abla imes ext{F}$ represent?

The curl of the vector field Signup and view all the answers

Study Notes