Let’s break down each part of your question:

a. Can a vector have nonzero magnitude if a component is zero? If no, why not? If yes, give an example.

Yes, a vector can have nonzero magnitude even if one of its components is zero.

Explanation:

• A vector is defined by its magnitude and direction, which are determined by its components along the coordinate axes.
• The magnitude of a vector is calculated using the components in each direction, but the presence of a zero component does not necessarily mean the entire vector’s magnitude is zero. It simply means that the vector has no contribution in that particular direction.

Example: Consider a vector $\mathbf{v}$ in 2-dimensional space given by $\mathbf{v}=⟨4,0⟩$.

• Here, the vector has a nonzero component in the x-direction (4) and a zero component in the y-direction (0).
• The magnitude of this vector is calculated as: Magnitude=(4)2+(0)2=16=4\text{Magnitude} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4Magnitude=(4)2+(0)2​=16​=4
• Thus, the vector $⟨4,0⟩$ has a nonzero magnitude (4) even though one of its components (y) is zero.

b. Can a vector have zero magnitude and a nonzero component? If no, why not? If yes, give an example.

No, a vector cannot have zero magnitude if it has a nonzero component.

Explanation:

• The magnitude of a vector is determined by all its components. If any component of the vector is nonzero, the magnitude will also be nonzero.
• The magnitude of a vector v=⟨vx,vy⟩\mathbf{v} = \langle v_x, v_y \ranglev=⟨vx​,vy​⟩ is given by: Magnitude=vx2+vy2\text{Magnitude} = \sqrt{v_x^2 + v_y^2}Magnitude=vx2​+vy2​​
• If either vxv_xvx​ or vyv_yvy​ is nonzero, then vx2+vy2v_x^2 + v_y^2vx2​+vy2​ is greater than zero, which means the magnitude of the vector is also greater than zero.

Example: Consider a vector $\mathbf{v}=⟨0,3⟩$.

• This vector has a nonzero component (3 in the y-direction) and zero in the x-direction.
• The magnitude of this vector is: Magnitude=(0)2+(3)2=9=3\text{Magnitude} = \sqrt{(0)^2 + (3)^2} = \sqrt{9} = 3Magnitude=(0)2+(3)2​=9​=3
• The vector $⟨0,3⟩$ has a nonzero magnitude (3) despite having a zero component in the x-direction.

At last:

• A vector can have a nonzero magnitude with a zero component.
• A vector cannot have zero magnitude if it has any nonzero components.