[amp_mcq option1=”\\[\\left[ {\\begin{array}{*{20}{c}} { – 6} \\\\ { – 3} \\\\ 6 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} 4 \\\\ { – 2} \\\\ 3 \\end{array}} \\right]\\]” option2=”\\[\\left[ {\\begin{array}{*{20}{c}} { – 4} \\\\ 2 \\\\ 4 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} 5 \\\\ 7 \\\\ { – 11} \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} 8 \\\\ 2 \\\\ { – 3} \\end{array}} \\right]\\]” option3=”\\[\\left[ {\\begin{array}{*{20}{c}} 6 \\\\ 7 \\\\ { – 1} \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} { – 3} \\\\ 2 \\\\ { – 2} \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ 9 \\\\ { – 4} \\end{array}} \\right]\\]” option4=”\\[\\left[ {\\begin{array}{*{20}{c}} 4 \\\\ 3 \\\\ {11} \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {31} \\\\ 3 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} 5 \\\\ 3 \\\\ 4 \\end{array}} \\right]\\]” correct=”option1″]<\/p>\n
The correct answer is $\\boxed{\\left[ {\\begin{array}{{20}{c}} { – 6} \\ { – 3} \\ 6 \\end{array}} \\right]\\left[ {\\begin{array}{<\/em>{20}{c}} 4 \\ { – 2} \\ 3 \\end{array}} \\right]}$.<\/p>\n To find an orthogonal set of vectors having a span that contains P, Q, R, we can use the Gram-Schmidt process. The Gram-Schmidt process is a method for orthonormalizing a set of vectors. It works by iteratively projecting each vector onto the space spanned by the previous vectors, and then normalizing the resulting vector.<\/p>\n In this case, we start with the vectors P, Q, and R. We then project P onto the space spanned by Q and R. This gives us the vector $\\mathbf{p}_1 = P – \\frac{\\langle P, Q \\rangle}{\\langle Q, Q \\rangle} Q – \\frac{\\langle P, R \\rangle}{\\langle R, R \\rangle} R$. We then normalize $\\mathbf{p}_1$ to get the vector $\\mathbf{v}_1$.<\/p>\n We then project Q onto the space spanned by $\\mathbf{v}_1$ and R. This gives us the vector $\\mathbf{q}_1 = Q – \\frac{\\langle Q, \\mathbf{v}_1 \\rangle}{\\langle \\mathbf{v}_1, \\mathbf{v}_1 \\rangle} \\mathbf{v}_1 – \\frac{\\langle Q, R \\rangle}{\\langle R, R \\rangle} R$. We then normalize $\\mathbf{q}_1$ to get the vector $\\mathbf{v}_2$.<\/p>\n We continue this process until we have projected all of the vectors onto the space spanned by the previous vectors. The resulting vectors are an orthogonal set of vectors having a span that contains P, Q, and R.<\/p>\n