Let veca=3hati-hatj+2hatk,$\vec{a}=3\hat{i}-\hat{j}+2\hat{k},$ vecb=vecatimes(hati-2hatk)$\vec{b}=\vec{a}\times(\hat{i}-2\hat{k})$ and vecc=vecbtimeshatk$\vec{c}=\vec{b}\times\hat{k}$. Then the projection of vecc-2hatj$\vec{c}-2\hat{j}$ on veca$\vec{a}$ is: [cite: 3358, 3359, 3364]

### Related Formula The scalar projection of vector vecv$\vec{v}$ onto vector vecw$\vec{w}$ is calculated as: textProjection = fracvecv cdot vecw|vecw|$$\text{Projection} = \frac{\vec{v} \cdot \vec{w}}{|\vec{w}|}$$ ### Step 1: Calculate vecb$\vec{b}$ Compute the cross product using standard matrix expansion [cite: 4013, 4015]: vecb = veca times (hati - 2hatk) = beginvmatrix hati & hatj & hatk \\ 3 & -1 & 2 \\ 1 & 0 & -2 endvmatrix$$\vec{b} = \vec{a} \times (\hat{i} - 2\hat{k}) = \begin{\vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -1 & 2 \\ 1 & 0 & -2 \end{\vmatrix}$$ vecb = hati(2 - 0) - hatj(-6 - 2) + hatk(0 - (-1)) = 2hati + 8hatj + hatk$$\vec{b} = \hat{i}(2 - 0) - \hat{j}(-6 - 2) + \hat{k}(0 - (-1)) = 2\hat{i} + 8\hat{j} + \hat{k}$$ ### Step 2: Calculate vecc$\vec{c}$ and vecc - 2hatj$\vec{c} - 2\hat{j}$ Perform the second cross product with unit vector hatk$\hat{k}$ [cite: 3359, 4016]: vecc = vecb times hatk = (2hati + 8hatj + hatk) times hatk = 2(hati times hatk) + 8(hatj times hatk) + vec0$$\vec{c} = \vec{b} \times \hat{k} = (2\hat{i} + 8\hat{j} + \hat{k}) \times \hat{k} = 2(\hat{i} \times \hat{k}) + 8(\hat{j} \times \hat{k}) + \vec{0}$$ vecc = 2(-hatj) + 8(hati) = 8hati - 2hatj$$\vec{c} = 2(-\hat{j}) + 8(\hat{i}) = 8\hat{i} - 2\hat{j}$$ Subtract 2hatj$2\hat{j}$ [cite: 3364, 4016]: vecc - 2hatj = (8hati - 2hatj) - 2hatj = 8hati - 4hatj$$\vec{c} - 2\hat{j} = (8\hat{i} - 2\hat{j}) - 2\hat{j} = 8\hat{i} - 4\hat{j}$$ ### Step 3: Compute the projection onto veca$\vec{a}$ Using the \dot product formula : textProjection = frac(vecc - 2hatj) cdot veca|veca| = fraclangle 8, -4, 0 rangle cdot langle 3, -1, 2 ranglesqrt3^2 + (-1)^2 + 2^2$$\text{Projection} = \frac{(\vec{c} - 2\hat{j}) \cdot \vec{a}}{|\vec{a}|} = \frac{\langle 8, -4, 0 \rangle \cdot \langle 3, -1, 2 \rangle}{\sqrt{3^2 + (-1)^2 + 2^2}}$$ [cite: 3358, 3995] textProjection = frac24 + 4 + 0sqrt9 + 1 + 4 = frac28sqrt14 = 2sqrt14$$\text{Projection} = \frac{24 + 4 + 0}{\sqrt{9 + 1 + 4}} = \frac{28}{\sqrt{14}} = 2\sqrt{14}$$ ### Pattern Recognition Keep cyclic unit cross products clear: hati times hatk = -hatj$\hat{i} \times \hat{k} = -\hat{j}$ and hatj times hatk = hati$\hat{j} \times \hat{k} = \hat{i}$. Missing a negative sign during basic cross multiplications ruins multi-step vector projections easily. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Vector Algebra