correct grouping and basic arithmetic operations earns the second mark M1."},{"type":"text","order":3,"title":"Calculate the final components","content":"Now compute the sum for each component:\n\n- **$\\mathbf{i}$-component:** $10 - 4 - 2 = 4$\n- **$\\mathbf{j}$-component:** $-3 - 6 + 12 = 3$\n- **$\\mathbf{k}$-component:** $2 + 5 - 3 = 4$\n\nPutting these together gives the final resultant force:\n\n$$\\mathbf{F}_{R} = 4\\mathbf{i} + 3\\mathbf{j} + 4\\mathbf{k}$$\n\nThis final vector earns you the final accuracy mark A1."}]},{"id":"9f61fcdd-1637-47c6-a68a-0e4d81c236d8:column_vector_addition","label":"Column Vector Notation","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = 10\\mathbf{i} - 3\\mathbf{j} + 2\\mathbf{k} \\implies \\htmlData{anchor=f1-vec}{\\begin{pmatrix} \\color{@sky}{10} \\\\ \\color{@amber}{-3} \\\\ \\color{@purple}{2} \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_2 = -4\\mathbf{i} - 6\\mathbf{j} + 5\\mathbf{k} \\implies \\htmlData{anchor=f2-vec}{\\begin{pmatrix} \\color{@sky}{-4} \\\\ \\color{@amber}{-6} \\\\ \\color{@purple}{5} \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_3 = -2\\mathbf{i} + 12\\mathbf{j} - 3\\mathbf{k} \\implies \\htmlData{anchor=f3-vec}{\\begin{pmatrix} \\color{@sky}{-2} \\\\ \\color{@amber}{12} \\\\ \\color{@purple}{-3} \\end{pmatrix}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"f1-vec"},{"color":"amber","style":"box","anchor":"f2-vec"},{"color":"purple","style":"box","anchor":"f3-vec"}],"connections":null},"order":0,"title":"Convert to column vectors","description":"To make adding the components easier and more organized, we can convert each force vector from $\\mathbf{i}, \\mathbf{j}, \\mathbf{k}$ unit notation into column vector form: $\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$."},{"type":"text","order":1,"title":"Sum the vectors","content":"The resultant force $\\mathbf{F}_{R}$ is the vector sum of all three forces: \n\n$$\\mathbf{F}_{R} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nSubstituting our column vectors gives us:\n\n$$\\mathbf{F}_{R} = \\begin{pmatrix} 10 \\\\ -3 \\\\ 2 \\end{pmatrix} + \\begin{pmatrix} -4 \\\\ -6 \\\\ 5 \\end{pmatrix} + \\begin{pmatrix} -2 \\\\ 12 \\\\ -3 \\end{pmatrix}$$\n\nThis step earns the first mark M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x\\text{-component}: \\color{@sky}{10} + (\\color{@sky}{-4}) + (\\color{@sky}{-2}) = \\htmlData{anchor=x-res}{\\color{@sky}{4}}$"},{"latex":"$$y\\text{-component}: \\color{@amber}{-3} + (\\color{@amber}{-6}) + \\color{@amber}{12} = \\htmlData{anchor=y-res}{\\color{@amber}{3}}$"},{"latex":"$$z\\text{-component}: \\color{@purple}{2} + \\color{@purple}{5} + (\\color{@purple}{-3}) = \\htmlData{anchor=z-res}{\\color{@purple}{4}}$"},{"latex":"$$\\mathbf{F}_R = \\begin{pmatrix} \\htmlData{anchor=x-col}{\\color{@sky}{4}} \\\\ \\htmlData{anchor=y-col}{\\color{@amber}{3}} \\\\ \\htmlData{anchor=z-col}{\\color{@purple}{4}} \\end{pmatrix}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x-res"},{"color":"amber","style":"text-color","anchor":"y-res"},{"color":"purple","style":"text-color","anchor":"z-res"}],"connections":[{"to":"x-col","from":"x-res","color":"sky","label":"x-sum","style":"dashed"},{"to":"y-col","from":"y-res","color":"amber","label":"y-sum","style":"dashed"},{"to":"z-col","from":"z-res","color":"purple","label":"z-sum","style":"dashed"}]},"order":2,"title":"Calculate the components","description":"Add the corresponding components from each of our three force vectors to find the components of the resultant force. This earns the second mark M1."},{"type":"text","order":3,"title":"Write the final vector","content":"Now, we convert the resultant column vector back to unit vector notation using $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$:\n\n$$\\mathbf{F}_{R} = 4\\mathbf{i} + 3\\mathbf{j} + 4\\mathbf{k}$$\n\nThis final answer completes the calculation and earns the final accuracy mark A1."}]}],"generatedAt":"2026-06-09T04:38:18.855Z","modelVersion":"gem

1 Pure Mathematics

Question 61

1 Pure Mathematics Questions

1 Pure Mathematics

Question 61

1 Pure Mathematics Questions