Find the scalar projection of the line segment joining the points $\mathrm{A}(2,-1,4)$ and $\mathrm{B}(5,3,1)$ onto the line with direction ratios $2,3,-6$.

Question

Aicharya · Accepted Answer

To find the scalar projection, we use the formula: $\text { Scalar Projection }=\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$
Where: - $\mathbf{a}$ is the vector from A to B . - bis the direction vector of the line. 1. Find vector a: $\mathbf{a}=B-A=(5-2) i+(3-(-1)) j+(1-4) k=3 i+4 j-3 k$ 2. Find vector $\mathbf{b}$ : $\mathbf{b}=2 i+3 j-6 k$ 3. Compute the dot product a b: $\mathbf{a} \cdot \mathbf{b}=(3)(2)+(4)(3)+(-3)(-6)=6+12+18=36$ 4. Find the magnitude of b : $|\mathbf{b}|=\sqrt{2^2+3^2+(-6)^2}=\sqrt{4+9+36}=\sqrt{49}=7$ 5. Calculate the scalar projection: $\text { Scalar Projection }=\frac{36}{7}$

Find the scalar projection of the line segment joining the points $\mathrm{A}(2,-1,4)$ and $\mathrm{B}(5,3…

Solution