Question Find the value of $c$ if the vectors $v_1 = [1, 0, 0]$ , $v_2 = [1, 1, 0]$ , and $v_3 = [1, -1, c]$ are linearly dependent.

Studdy Solution

STEP 1

Assumptions
1. The vectors $v_{1}$ , $v_{2}$ , and $v_{3}$ are given.
2. The vectors are linearly dependent.

STEP 2

By definition, vectors are linearly dependent if there exist scalars, not all zero, such that a linear combination of the vectors equals the zero vector. We can express this as:
$a v_{1} + b v_{2} + c v_{3} = \mathbf{0}$
where $a$ , $b$ , and $c$ are scalars and $\mathbf{0}$ is the zero vector.

STEP 3

Substitute the given vectors into the linear combination equation.
$a \left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right] + b \left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right] + c \left[\begin{array}{c}1 \\ -1 \\ c\end{array}\right] = \left[\begin{array}{c}0 \\ 0 \\ 0\end{array}\right]$

STEP 4

Expand the equation to get a system of linear equations.
$\begin{cases} a + b + c = 0 \\ 0 + b - c = 0 \\ 0 + 0 + c^2 = 0 \end{cases}$

STEP 5

From the third equation $c^2 = 0$ , we can immediately find the value of $c$ .
$c^2 = 0$

STEP 6

Solve for $c$ .
$c = 0$
Since the vectors are linearly dependent, the value of $c$ that satisfies the condition is $c=0$ .

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Question Find the value of $c$ if the vectors $v_1 = [1, 0, 0]$ , $v_2 = [1, 1, 0]$ , and $v_3 = [1, -1, c]$ are linearly dependent.

Studdy Solution

STEP 1

Assumptions
1. The vectors $v_{1}$ , $v_{2}$ , and $v_{3}$ are given.
2. The vectors are linearly dependent.

STEP 2

By definition, vectors are linearly dependent if there exist scalars, not all zero, such that a linear combination of the vectors equals the zero vector. We can express this as:
$a v_{1} + b v_{2} + c v_{3} = \mathbf{0}$
where $a$ , $b$ , and $c$ are scalars and $\mathbf{0}$ is the zero vector.

STEP 3

Substitute the given vectors into the linear combination equation.
$a \left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right] + b \left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right] + c \left[\begin{array}{c}1 \\ -1 \\ c\end{array}\right] = \left[\begin{array}{c}0 \\ 0 \\ 0\end{array}\right]$

STEP 4

Expand the equation to get a system of linear equations.
$\begin{cases} a + b + c = 0 \\ 0 + b - c = 0 \\ 0 + 0 + c^2 = 0 \end{cases}$

STEP 5

From the third equation $c^2 = 0$ , we can immediately find the value of $c$ .
$c^2 = 0$

STEP 6

Solve for $c$ .
$c = 0$
Since the vectors are linearly dependent, the value of $c$ that satisfies the condition is $c=0$ .

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Question Find the value of ccc if the vectors v1=[1,0,0]v_1 = [1, 0, 0]v1​=[1,0,0], v2=[1,1,0]v_2 = [1, 1, 0]v2​=[1,1,0], and v3=[1,−1,c]v_3 = [1, -1, c]v3​=[1,−1,c] are linearly dependent.

Studdy Solution

STEP 1

Assumptions1. The vectors v1v_{1}v1​, v2v_{2}v2​, and v3v_{3}v3​ are given.2. The vectors are linearly dependent.

STEP 2

STEP 3

STEP 4

Expand the equation to get a system of linear equations.{a+b+c=00+b−c=00+0+c2=0\begin{cases} a + b + c = 0 \\ 0 + b - c = 0 \\ 0 + 0 + c^2 = 0 \end{cases}⎩⎨⎧​a+b+c=00+b−c=00+0+c2=0​

STEP 5

From the third equation c2=0c^2 = 0c2=0, we can immediately find the value of ccc.c2=0c^2 = 0c2=0

STEP 6

Solve for ccc.c=0c = 0c=0Since the vectors are linearly dependent, the value of ccc that satisfies the condition is c=0c=0c=0.

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Question Find the value of $c$ if the vectors $v_1 = [1, 0, 0]$ , $v_2 = [1, 1, 0]$ , and $v_3 = [1, -1, c]$ are linearly dependent.

Assumptions
1. The vectors $v_{1}$ , $v_{2}$ , and $v_{3}$ are given.
2. The vectors are linearly dependent.

Expand the equation to get a system of linear equations.
$\begin{cases} a + b + c = 0 \\ 0 + b - c = 0 \\ 0 + 0 + c^2 = 0 \end{cases}$

From the third equation $c^2 = 0$ , we can immediately find the value of $c$ .
$c^2 = 0$

Solve for $c$ .
$c = 0$
Since the vectors are linearly dependent, the value of $c$ that satisfies the condition is $c=0$ .