Questions: Find the value(s) of h for which the vectors are linearly dependent. Justify your answer. [2, -2, 6] · [6, -8, 17] · [-3, 3, h] The value(s) of h which makes the vectors linearly dependent is(are) because this will cause to be a variable.

Find the value(s) of h for which the vectors are linearly dependent. Justify your answer.

[2, -2, 6] · [6, -8, 17] · [-3, 3, h]

The value(s) of h which makes the vectors linearly dependent is(are) because this will cause to be a variable.

Transcript text: Find the value(s) of $h$ for which the vectors are linearly dependent. Justify your answer. \[ \left[\begin{array}{r} 2 \\ -2 \\ 6 \end{array}\right] \cdot\left[\begin{array}{r} 6 \\ -8 \\ 17 \end{array}\right] \cdot\left[\begin{array}{r} -3 \\ 3 \\ h \end{array}\right] \] The value(s) of h which makes the vectors linearly dependent is(are) $\square$ because this will cause $\square$ to be a $\square$ variable. (Use a comma to separate answers as needed.)

Solution

Solution Steps

Step 1: Form the Matrix

We start by forming a matrix $ A $ using the given vectors as columns: \[ A = \begin{bmatrix} 2 & 6 & -3 \\ -2 & -8 & 3 \\ 6 & 17 & h \end{bmatrix} \]

Step 2: Calculate the Determinant

To find the values of $ h $ for which the vectors are linearly dependent, we need to calculate the determinant of matrix $ A $ and set it equal to zero: \[ \text{Det}(A) = \left|\begin{matrix} 2 & 6 & -3 \\ -2 & -8 & 3 \\ 6 & 17 & h \end{matrix}\right| \]

Step 3: Evaluate Determinant for Different Values of $ h $

We evaluate the determinant for various integer values of $ h $ from -10 to 10. The calculations yield the following results:

For $ h = -10 $: \[ \text{Det}(A) = 4.00 \]
For $ h = -9 $: \[ \text{Det}(A) = 0.00 \]
For $ h = -8 $: \[ \text{Det}(A) = -4.00 \]
For $ h = -7 $: \[ \text{Det}(A) = -8.00 \]

Step 4: Identify Linear Dependence

The vectors are linearly dependent when the determinant is equal to zero. From our calculations, we find that: \[ \text{Det}(A) = 0 \quad \text{when} \quad h = -9 \]

Step 5: Conclusion

The value of $ h $ that makes the vectors linearly dependent is: \[ h = -9 \]