Vector Algebra
Linear dependence of vectors
Grade 12
Question:
<p>It is given that \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are linearly dependent, where \(\vec{a} = \hat{i}+\hat{j}+\hat{k}\), \(\vec{b} = 4\hat{i}+3\hat{j}+4\hat{k}\), \(\vec{c} = \hat{i}+\alpha\hat{j}+\beta\hat{k}\). Find \(\beta\).</p>
Step-by-Step Solution
Key Concept: Three vectors are linearly dependent if and only if their scalar triple product equals zero, which means the determinant of the matrix formed by these vectors as rows (or columns) is zero.
Step 1: For vectors a , b , and c to be linearly dependent, the scalar triple product must equal zero: $\begin{vmatrix} 1 & 1 & 1 \\ 4 & 3 & 4 \\ 1 & \alpha & \beta \end{vmatrix} = 0$ Step 2: Expand the determinant along the first row: $1\begin{vmatrix} 3 & 4 \\ \alpha & \beta \end{vmatrix} - 1\begin{vmatrix} 4 & 4 \\ 1 & \beta \end{vmatrix} + 1\begin{vmatrix} 4 & 3 \\ 1 & \alpha \end{vmatrix} = 0$ Step 3: Calculate each 2×2 determinant: $1(3\beta - 4\alpha) - 1(4\beta - 4) + 1(4\alpha - 3) = 0$ $3\beta - 4\alpha - 4\beta + 4 + 4\alpha - 3 = 0$ $-\beta + 1 = 0$ Step 4: Solve for β: $\beta = 1$ ∴ Answer: 1
Correct Answer: 1