Question

Three honey bees were found flying along the vectors $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 4\hat{j} - 2\hat{k}$ and $\vec{c} = 3\hat{i} + 2\hat{k}$ respectively. Find the scalar value of $\lambda$ such that the path represented by the vector $\vec{a} + \lambda \vec{b}$ is perfectly perpendicular to the vector $\vec{c}$.

Hint:

When computing dot products where one component is missing (like $\hat{j}$ in vector $\vec{c}$), that entire middle term becomes zero automatically. Pay close attention to missing components to avoid calculation errors!

Answer 1

Concept: In vector algebra, two non-zero vectors $\vec{u}$ and $\vec{v}$ are mathematically defined to be perpendicular (orthogonal) to each other if and only if their vector dot product (scalar product) is exactly equal to zero. That is: \[ \vec{u} \cdot \vec{v} = 0 \] The dot product of two vectors represented in Cartesian component form $\vec{u} = u_1\hat{i} + u_2\hat{j} + u_3\hat{k}$ and $\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$ is computed as: \[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \] We will form a linear equation in terms of $\lambda$ using this condition and solve for it.

Step 1: Constructing the composite vector $\vec{a} + \lambda \vec{b}$ in component form.
Let us write out the full component expressions for vectors $\vec{a}$ and $\vec{b}$: \[ \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \] \[ \vec{b} = 0\hat{i} + 4\hat{j} - 2\hat{k} \] Now, let us perform scalar multiplication of $\lambda$ with vector $\vec{b}$: \[ \lambda\vec{b} = \lambda(4\hat{j} - 2\hat{k}) = 0\hat{i} + 4\lambda\hat{j} - 2\lambda\hat{k} \] Next, add vector $\vec{a}$ and vector $\lambda\vec{b}$ together by grouping their respective $\hat{i}$, $\hat{j}$, and $\hat{k}$ directional unit components: \[ \vec{a} + \lambda\vec{b} = (2 + 0)\hat{i} + (-3 + 4\lambda)\hat{j} + (1 - 2\lambda)\hat{k} \] \[ \vec{a} + \lambda\vec{b} = 2\hat{i} + (4\lambda - 3)\hat{j} + (1 - 2\lambda)\hat{k} \]

Step 2: Applying the orthogonality condition with vector $\vec{c}$.
The vector representing the third path is given as: \[ \vec{c} = 3\hat{i} + 0\hat{j} + 2\hat{k} \] Since the question states that $(\vec{a} + \lambda\vec{b})$ is perpendicular to $\vec{c}$, their scalar dot product must be equal to zero: \[ (\vec{a} + \lambda\vec{b}) \cdot \vec{c} = 0 \] Substitute the derived component values into this dot product equation: \[ \left[ 2\hat{i} + (4\lambda - 3)\hat{j} + (1 - 2\lambda)\hat{k} \right] \cdot \left[ 3\hat{i} + 0\hat{j} + 2\hat{k} \right] = 0 \] Multiply corresponding directional coefficients together: \[ (2)(3) + (4\lambda - 3)(0) + (1 - 2\lambda)(2) = 0 \]

Step 3: Solving the resulting algebraic equation to determine $\lambda$.
Let us expand and simplify the expression: \[ 6 + 0 + 2(1) - 2(2\lambda) = 0 \] \[ 6 + 2 - 4\lambda = 0 \] \[ 8 - 4\lambda = 0 \] Isolating the variable term containing $\lambda$: \[ 4\lambda = 8 \implies \lambda = \frac{8}{4} \implies \lambda = 2 \] Thus, the value of the scalar parameter $\lambda$ must be exactly $2$.