Math Problem Statement
you are given vector AB = (0.82, -5.36, -1) and vector AC = (4.75, -3.41, -1). What is the cross product of AB x AC?
Solution
To calculate the cross product of two vectors and , we use the formula:
\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0.82 & -5.36 & -1 \\ 4.75 & -3.41 & -1 \end{vmatrix}$$ Expanding the determinant: $$\mathbf{AB} \times \mathbf{AC} = \mathbf{i} \begin{vmatrix} -5.36 & -1 \\ -3.41 & -1 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 0.82 & -1 \\ 4.75 & -1 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 0.82 & -5.36 \\ 4.75 & -3.41 \end{vmatrix}$$ ### Step-by-Step Calculation: 1. Compute $$\mathbf{i}$$-component: $$\begin{vmatrix} -5.36 & -1 \\ -3.41 & -1 \end{vmatrix} = (-5.36)(-1) - (-3.41)(-1) = 5.36 - 3.41 = 1.95$$ 2. Compute $$\mathbf{j}$$-component: $$\begin{vmatrix} 0.82 & -1 \\ 4.75 & -1 \end{vmatrix} = (0.82)(-1) - (4.75)(-1) = -0.82 + 4.75 = 3.93$$ 3. Compute $$\mathbf{k}$$-component: $$\begin{vmatrix} 0.82 & -5.36 \\ 4.75 & -3.41 \end{vmatrix} = (0.82)(-3.41) - (4.75)(-5.36) = -2.7962 + 25.46 = 22.6638$$ ### Combine Results: $$\mathbf{AB} \times \mathbf{AC} = 1.95\mathbf{i} - 3.93\mathbf{j} + 22.6638\mathbf{k}$$ In vector notation: $$\mathbf{AB} \times \mathbf{AC} = (1.95, -3.93, 22.6638)$$ --- Would you like a more detailed explanation of any step or its geometric interpretation? 😊 --- ### Related Questions: 1. What is the magnitude of the resulting cross product vector? 2. How does the cross product relate to the area of the parallelogram spanned by AB and AC? 3. How do we find the angle between the two vectors using the cross product? 4. What is the unit vector in the direction of the cross product? 5. How can we verify the orthogonality of the cross product to the original vectors? ### Tip: The cross product of two vectors is always perpendicular to the plane formed by the two vectors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Operations
Cross Product
Linear Algebra
Formulas
Cross product formula: \( \mathbf{AB} \times \mathbf{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ AB_x & AB_y & AB_z \\ AC_x & AC_y & AC_z \end{vmatrix} \)
Determinant calculation for 2x2 matrices: \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \)
Theorems
Cross Product Properties
Orthogonality of Cross Product
Suitable Grade Level
Grades 10-12
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