[amp_mcq option1=”\\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} – \\overline {\\rm{b}} } \\right) \\cdot \\left( {\\overline {\\rm{a}} – \\overline {\\rm{c}} } \\right)\\]” option2=”\\[\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} – \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} – \\overline {\\rm{c}} } \\right)} \\right|\\]” option3=”\\[\\frac{1}{2}\\left| {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} \\times \\overline {\\rm{c}} } \\right|\\]” option4=”\\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} } \\right) \\cdot \\overline {\\rm{c}} \\]” correct=”option1″]<\/p>\n
The correct answer is $\\boxed{\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} – \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} – \\overline {\\rm{c}} } \\right)} \\right|}$.<\/p>\n
The area of a triangle formed by the tips of vectors $\\overline {\\rm{a}} {\\rm{,}}\\,\\overline {\\rm{b}} ] and [\\overline {\\rm{c}} ] is given by the formula<\/p>\n
$$\\text{Area} = \\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} – \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} – \\overline {\\rm{c}} } \\right)} \\right|$$<\/p>\n
where $\\times$ denotes the vector cross product.<\/p>\n
The vector cross product is a binary operation on two vectors that produces a vector perpendicular to both of the original vectors. The magnitude of the vector cross product is equal to the area of the parallelogram spanned by the two vectors, and the direction of the vector cross product is given by the right-hand rule.<\/p>\n
In this case, the vectors $\\overline {\\rm{a}} – \\overline {\\rm{b}} $ and $\\overline {\\rm{a}} – \\overline {\\rm{c}} $ are adjacent sides of the triangle, so the area of the triangle is given by the formula<\/p>\n
$$\\text{Area} = \\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} – \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} – \\overline {\\rm{c}} } \\right)} \\right|$$<\/p>\n
The other options are incorrect. Option A is the formula for the scalar product of two vectors. Option B is the formula for the magnitude of the vector cross product. Option C is the formula for the volume of a parallelepiped spanned by three vectors.<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”\\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} – \\overline {\\rm{b}} } \\right) \\cdot \\left( {\\overline {\\rm{a}} – \\overline {\\rm{c}} } \\right)\\]” option2=”\\[\\frac{1}{2}\\left| {\\left( {\\overline {\\rm{a}} – \\overline {\\rm{b}} } \\right) \\times \\left( {\\overline {\\rm{a}} – \\overline {\\rm{c}} } \\right)} \\right|\\]” option3=”\\[\\frac{1}{2}\\left| {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} \\times \\overline {\\rm{c}} } \\right|\\]” option4=”\\[\\frac{1}{2}\\left( {\\overline {\\rm{a}} \\times \\overline {\\rm{b}} } … <\/p>\n