Vector Cross Product and Applications

• |\vec{A} \times \vec{B}|

• |\vec{A} + \vec{B}|

• |\vec{A}| \times |\vec{B}|

• |\vec{A} + \vec{B}|

What is the formula for torque in terms of vectors r and F ?

• \vec{\tau} = \vec{r} \times \vec{F}

• \vec{\tau} = \vec{F} \times \vec{r}

Which of the following is a unit vector that lies in the plane 2x+3y-5z=5 and is parallel to the vector x̂-ĵ+k̂?

• (1/√3)(x̂-ĵ+k̂)

• (1/√2)(x̂+ĵ)

• (1/√6)(2x̂+3ĵ-5k̂)

• (1/2)(x̂+k̂)

What is the gradient of a scalar function φ(x, y, z) as defined in the notes?

• The gradient of a scalar function φ(x, y, z) is defined as ∇φ = i ∂φ/∂x + j ∂φ/∂y + k ∂φ/∂z.

• The gradient of a scalar function φ(x, y, z) is defined as ∇φ = i ∂φ/∂y + j ∂φ/∂z + k ∂φ/∂x.

• The gradient of a scalar function φ(x, y, z) is defined as ∇φ = i ∂φ/∂z + j ∂φ/∂x + k ∂φ/∂y.

• The gradient of a scalar function φ(x, y, z) is defined as ∇φ = i φ + j φ + k φ.

Is the gradient of a scalar function a vector or a scalar?

• Vector

• Scalar

The magnitude of ∇φ is equal to the maximum rate of change of φ with respect to the space variables and has the direction of that change.

• True

• False

What is the formula for the divergence of a vector A⃗ ?

• The divergence of a vector A⃗ is defined as div A⃗ = ∇·A⃗ = ∂A_x/∂x + ∂A_y/∂y + ∂A_z/∂z.

• The divergence of a vector A⃗ is defined as div A⃗ = ∇×A⃗ = ∂A_x/∂y + ∂A_y/∂z + ∂A_z/∂x.

• The divergence of a vector A⃗ is defined as div A⃗ = ∇·A⃗ = ∂A_x/∂y + ∂A_y/∂z + ∂A_z/∂x.

• The divergence of a vector A⃗ is defined as div A⃗ = ∇·A⃗ = ∂A_x/∂z + ∂A_y/∂x + ∂A_z/∂y.

Divergence of a vector A⃗ is defined as the scalar product of the vector differential operator ∇ and A⃗.

• True

• False