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A gradient vector is a fundamental tool in multivariable calculus used to describe how a function changes at any given point in space. For any function of three variables, such as \( f(x, y, z) \), the gradient vector, denoted as \( abla f \), is computed by taking the partial derivatives of \( f \) with respect to \( x \), \( y \), and \( z \):

• \( \frac{\partial f}{\partial x} \)
• \( \frac{\partial f}{\partial y} \)
• \( \frac{\partial f}{\partial z} \)

For the surface described by \( f(x, y, z) = x^2 + 2y + 2z - 4 \), the gradient vector becomes \( abla f = (2x, 2, 2) \). Meanwhile, for a simpler linear surface like \( g(x, y, z) = y - 1 \), the gradient is constant: \( abla g = (0, 1, 0) \).

These vectors provide the direction of steepest ascent on the surface, and by evaluating them at specific points such as \((1, 1, 1/2)\), you get \( abla f = (2, 2, 2) \) and \( abla g = (0, 1, 0) \). Gradient vectors are vital in finding tangent lines and planes to surfaces, as they highlight crucial directional information.

The cross product is a mathematical operation used to find a vector that is perpendicular to two given vectors in three-dimensional space, which is especially useful in physics and engineering applications. For vectors \( \mathbf{A} = (a_1, a_2, a_3) \) and \( \mathbf{B} = (b_1, b_2, b_3) \), the cross product, \( \mathbf{A} \times \mathbf{B} \), is defined as:

• \((a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\)

In the context of finding a tangent line, the cross product of the gradient vectors \( abla f \) and \( abla g \) gives us the direction of the tangent line.

When you take the cross product of \( (2, 2, 2) \) and \( (0, 1, 0) \), you get \( ( -2, 0, 2 ) \). This resulting vector represents the direction in which the tangent line extends. The cross product works efficiently because it combines information from both surfaces to find the common tangent direction.

Parametric equations are a way of defining a line or curve by expressing the coordinates as functions of a parameter, often denoted by \( t \). This is particularly useful for describing motion or lines in space, where each equation represents a dimension:

• For the \( x \)-coordinate: \( x(t) = x_0 + at \)
• For the \( y \)-coordinate: \( y(t) = y_0 + bt \)
• For the \( z \)-coordinate: \( z(t) = z_0 + ct \)

In the case of the tangent line problem, given the starting point \((1, 1, 1/2)\) and the direction vector \(( -2, 0, 2 )\), the parametric equations become:

\[x = 1 - 2t,y = 1,z = \frac{1}{2} + 2t\]These equations describe a line in space beginning at the point \((1, 1, 1/2)\) and moving in the direction provided by the cross product. Parametric equations are versatile and can clearly convey how points on a line change with respect to the parameter \( t \).