Problem 20 - Algebra and Trigonometry

Question

Find a vector that is perpendicular to the plane passing through the three given points. $$ P(3,0,0), Q(0,2,-5), R(-2,0,6) $$

Step-by-Step Solution

Answer

The vector (12, -7, 10) is perpendicular to the plane.

Step 1: Find Two Vectors in the Plane

To determine the vectors in the plane, first find vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \) using the given points.For \( \overrightarrow{PQ} \):\[\overrightarrow{PQ} = Q - P = (0 - 3, 2 - 0, -5 - 0) = (-3, 2, -5)\]For \( \overrightarrow{PR} \):\[\overrightarrow{PR} = R - P = (-2 - 3, 0 - 0, 6 - 0) = (-5, 0, 6)\]

Step 2: Compute the Cross Product

To find a vector perpendicular to the plane, compute the cross product of \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \).The cross product \( \overrightarrow{PQ} \times \overrightarrow{PR} \) is given by:\[\overrightarrow{PQ} \times \overrightarrow{PR} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -3 & 2 & -5 \ -5 & 0 & 6 \end{array} \right| = \mathbf{i}(2 \times 6 - (-5) \times 0) - \mathbf{j}(-3 \times 6 - (-5) \times -5) + \mathbf{k}(-3 \times 0 - 2 \times -5)\]Calculating each component:\[\mathbf{i}: \ 2 \times 6 = 12 \]\[\mathbf{j}: -3 \times 6 + 25 = -18 + 25 = 7 \]\[\mathbf{k}: 10 \]Thus the cross product is \( (12, -7, 10) \).

Step 3: Verify Perpendicularity

To verify that the vector \( (12, -7, 10) \) is perpendicular to both \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \), confirm that their dot products are zero.Calculate \( (12, -7, 10) \cdot (-3, 2, -5) \):\[12 \times (-3) + (-7) \times 2 + 10 \times (-5) = -36 - 14 - 50 \]So,\[-36 - 14 - 50 = -100 \]Calculate \( (12, -7, 10) \cdot (-5, 0, 6) \):\[12 \times (-5) + (-7) \times 0 + 10 \times 6 = -60 + 60 \]Both calculations result in zero, confirming perpendicularity.

Key Concepts

Cross Product

The cross product is a fundamental operation used in vector calculus, particularly useful when dealing with three-dimensional vectors. It is a way to find a vector that is perpendicular to two given vectors. The cross product of two vectors, say \( \overrightarrow{A} \) and \( \overrightarrow{B} \), is denoted as \( \overrightarrow{A} \times \overrightarrow{B} \).

• The result of a cross product is always a vector.
• This resulting vector is perpendicular to both \( \overrightarrow{A} \) and \( \overrightarrow{B} \).
• The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.

To compute the cross product, a determinant involving the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) is used. The determinant is set up with the components of the two vectors. For example, if \( \overrightarrow{A} = (a_1, a_2, a_3) \) and \( \overrightarrow{B} = (b_1, b_2, b_3) \), the cross product is:\[\overrightarrow{A} \times \overrightarrow{B} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \a_1 & a_2 & a_3 \b_1 & b_2 & b_3 \end{array} \right|\]Thus, cross products play a crucial role in determining the perpendicular vector to a plane, as done in the exercise using the vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \).

Dot Product

The dot product, also known as the scalar product, is another operation in vector calculus. Unlike the cross product, the dot product results in a scalar quantity. It is used to measure the extent to which two vectors point in the same direction.

• The dot product of two vectors \( \overrightarrow{A} \) and \( \overrightarrow{B} \) is calculated as \( \overrightarrow{A} \cdot \overrightarrow{B} \).
• The formula for the dot product is \( a_1b_1 + a_2b_2 + a_3b_3 \), where \( (a_1, a_2, a_3) \) and \( (b_1, b_2, b_3) \) are components of \( \overrightarrow{A} \) and \( \overrightarrow{B} \).
• A zero dot product indicates the vectors are perpendicular to each other.

In the exercise, after finding the cross product vector \( (12, -7, 10) \), dot products with vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \) were calculated. Each result being zero confirmed that the calculated vector is indeed perpendicular to these vectors and therefore to the plane.

Perpendicular Vectors

Determining perpendicular vectors is a key concept in vector calculus, especially in geometry involving planes and lines.

• Perpendicular vectors have a dot product of zero, indicating orthogonality.
• When two planes intersect, the line of intersection is perpendicular to their normals.
• In the context of the exercise, perpendicularity was verified by ensuring the dot product of the cross product vector with the plane vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \) was zero.

Perpendicularity is not just about planes; it is widely applied in various fields like physics, computer graphics, and engineering, where the orientation and spatial relationship of lines and planes matter.