Question

Let \( \vec{p} = 2\hat{i} - 3\hat{j} - \hat{k} \), \( \vec{q} = -3\hat{i} + 4\hat{j} + \hat{k} \) and \( \vec{r} = \hat{i} + \hat{j} + 2\hat{k} \). Express \( \vec{r} \) in the form of \( \vec{r} = \lambda \vec{p} + \mu \vec{q} \) and hence find the values of \( \lambda \) and \( \mu \).

Hint:

When expressing a vector as a linear combination of other vectors, the problem reduces to solving a system of linear equations for the scalar coefficients. If the vectors are in three dimensions, you will typically get a system of three equations with three unknowns. Ensure your algebraic manipulations are accurate to find the correct values of the scalars.

Answer 1

We are given the vectors: $$\vec{p} = 2\hat{i} - 3\hat{j} - \hat{k}$$ $$\vec{q} = -3\hat{i} + 4\hat{j} + \hat{k}$$ $$\vec{r} = \hat{i} + \hat{j} + 2\hat{k}$$ We need to find scalars \( \lambda \) and \( \mu \) such that \( \vec{r} = \lambda \vec{p} + \mu \vec{q} \). Substituting the vectors: $$\hat{i} + \hat{j} + 2\hat{k} = \lambda (2\hat{i} - 3\hat{j} - \hat{k}) + \mu (-3\hat{i} + 4\hat{j} + \hat{k})$$ $$\hat{i} + \hat{j} + 2\hat{k} = (2\lambda - 3\mu)\hat{i} + (-3\lambda + 4\mu)\hat{j} + (-\lambda + \mu)\hat{k}$$ Equating the coefficients of \( \hat{i}, \hat{j}, \) and \( \hat{k} \), we get the following system of linear equations: $$2\lambda - 3\mu = 1 \quad (1)$$ $$-3\lambda + 4\mu = 1 \quad (2)$$ $$-\lambda + \mu = 2 \quad (3)$$ From equation (3), we can express \( \mu \) in terms of \( \lambda \): $$\mu = \lambda + 2$$ Substitute this into equation (1): $$2\lambda - 3(\lambda + 2) = 1$$ $$2\lambda - 3\lambda - 6 = 1$$ $$-\lambda = 7$$ $$\lambda = -7$$ Now, substitute the value of \( \lambda \) back into the expression for \( \mu \): $$\mu = -7 + 2$$ $$\mu = -5$$ To verify, substitute \( \lambda = -7 \) and \( \mu = -5 \) into equation (2): $$-3(-7) + 4(-5) = 21 - 20 = 1$$ The values satisfy all three equations. Thus, \( \vec{r} = -7\vec{p} - 5\vec{q} \), with \( \lambda = -7 \) and \( \mu = -5 \).