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Find the value(s) of h for which the vectors are linearly dependent. Justify your answer. 2 -2 4 2 7 - 3 3 h The value(s) of h which makes the vectors linearly dependent is(are) because this will cause variable. (Use a comma to separate answers as needed.) to be a

Find the value(s) of h for which the vectors are linearly dependent. Justify your answer. 2 -2 4 2 7 - 3 3 h The value(s) of h which makes the vectors linearly dependent is(are) because this will cause variable. (Use a comma to separate answers as needed.) to be a

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 21E
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### Determining Values for Linear Dependence #### Problem Statement Find the value(s) of \( h \) for which the vectors are linearly dependent. Justify your answer. \[ \begin{bmatrix} 2 \\ -2 \\ -4 \end{bmatrix} , \begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix} , \begin{bmatrix} -3 \\ 3 \\ h \end{bmatrix} \] Please identify the value(s) of \( h \) that make the vectors linearly dependent, and explain why. #### Solution Process The value(s) of \( h \) which make the vectors linearly dependent is (are) \(\boxed{\phantom{x}}\) because this will cause \(\boxed{\phantom{x}}\) to be a \(\boxed{\phantom{x}}\) variable. (Use a comma to separate answers as needed.) ### Explanation To determine the values of \( h \) for which the vectors are linearly dependent, we need to check when the determinant of the matrix formed by these vectors is zero. Linear dependence among the vectors implies that there is some non-trivial linear combination of these vectors that results in the zero vector. #### Steps to Solve: 1. Arrange the vectors as columns of a matrix. 2. Compute the determinant of the matrix. 3. Set the determinant equal to zero and solve for \( h \). ### Determinant Calculation: Given the matrix: \[ \begin{bmatrix} 2 & -4 & -3 \\ -2 & 2 & 3 \\ -4 & 7 & h \end{bmatrix} \] To find the value of \( h \) that makes the determinant of this matrix zero, compute the determinant and set it equal to zero. After determining the determinant, solve for \( h \). If you need further assistance with the determinant calculation or any specific steps, please refer to the resources on matrix determinants.
Transcribed Image Text:### Determining Values for Linear Dependence #### Problem Statement Find the value(s) of \( h \) for which the vectors are linearly dependent. Justify your answer. \[ \begin{bmatrix} 2 \\ -2 \\ -4 \end{bmatrix} , \begin{bmatrix} -4 \\ 2 \\ 7 \end{bmatrix} , \begin{bmatrix} -3 \\ 3 \\ h \end{bmatrix} \] Please identify the value(s) of \( h \) that make the vectors linearly dependent, and explain why. #### Solution Process The value(s) of \( h \) which make the vectors linearly dependent is (are) \(\boxed{\phantom{x}}\) because this will cause \(\boxed{\phantom{x}}\) to be a \(\boxed{\phantom{x}}\) variable. (Use a comma to separate answers as needed.) ### Explanation To determine the values of \( h \) for which the vectors are linearly dependent, we need to check when the determinant of the matrix formed by these vectors is zero. Linear dependence among the vectors implies that there is some non-trivial linear combination of these vectors that results in the zero vector. #### Steps to Solve: 1. Arrange the vectors as columns of a matrix. 2. Compute the determinant of the matrix. 3. Set the determinant equal to zero and solve for \( h \). ### Determinant Calculation: Given the matrix: \[ \begin{bmatrix} 2 & -4 & -3 \\ -2 & 2 & 3 \\ -4 & 7 & h \end{bmatrix} \] To find the value of \( h \) that makes the determinant of this matrix zero, compute the determinant and set it equal to zero. After determining the determinant, solve for \( h \). If you need further assistance with the determinant calculation or any specific steps, please refer to the resources on matrix determinants.
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