","comment":{"@type":"Comment","text":"From the trigonometric relations, $\\tan \\theta =\\dfrac{Perpendicular}{Base}$.From here calculate the angle."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 30$${}^\\circ $$25’","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" 60$${}^\\circ $$2’","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" 45$${}^\\circ $$13’","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 36$${}^\\circ $$51’","position":2,"answerExplanation":{"@type":"Comment","text":" $$\\text{From trigonometric relations, we know that,}\\\\ \\tan \\theta =\\dfrac{Perpendicular}{Base}\\\\ \\text{So,}\\\\ \\tan \\theta =\\dfrac{3}{4} \\\\ \\Rightarrow\\theta ={{\\tan }^{-1}}\\left( \\dfrac{3}{4} \\right) \\\\ \\therefore\\theta =36{}^\\circ 51'$$ ","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Vector components Quiz 1","text":" Consider a vector whose length is 5 cm and it is oriented $$30{}^\\circ $$ with the positive X-axis. Calculate it’s projections along the X and Y axis. $$ \\\\ $$ ","comment":{"@type":"Comment","text":" From the trigonometric relations, $\\sin \\theta =\\dfrac{Perpendicular}{Hypotenuse}$ and $\\cos \\theta =\\dfrac{Base}{Hypotenuse}$. Use these relations to find the answer. "},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" x=2.5, y=4.3","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" x=3.9, y=4.9","position":2},{"@type":"Answer","encodingFormat":"text/html","text":" x=4.9, y=3.9","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" x=4.3, y=2.5","position":1,"answerExplanation":{"@type":"Comment","text":" $$\\text{From the trigonometric relations, W.K.T}\\\\ \\sin 30{}^\\circ =\\dfrac{y}{5}\\\\ y=5\\sin 30{}^\\circ \\\\ y=2.5 \\\\ \\text{Similarly, }\\cos (30{}^\\circ )=\\dfrac{x}{5}\\\\ x=5\\cos (30{}^\\circ ) \\\\ \\therefore x=4.3 \\\\ \\text{So, the projections on x and y axis are,} x=4.3\\text{ and }y=2.5$$ ","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Vector components Quiz 1","text":" Calculate the projection of a vector along x-axis, whose length is 5 cm and it’s orientation with y-axis is $$60{}^\\circ $$. $$ \\\\ $$ ","comment":{"@type":"Comment","text":" The angle that it makes with x-axis is, $90{}^\\circ -60{}^\\circ =30{}^\\circ $.After finding the angle, use trigonometric ratios to find the answer. "},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 5.4","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" 3.3","position":2},{"@type":"Answer","encodingFormat":"text/html","text":" 6.4","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 4.3","position":0,"answerExplanation":{"@type":"Comment","text":" $$\\text{The angle that the vector makes with the x-axis is }90{}^\\circ -60{}^\\circ =30{}^\\circ\\\\ \\text{Now, }\\sin 30{}^\\circ =\\dfrac{y}{5}\\\\ y=5\\left( \\dfrac{1}{2} \\right) \\\\ y=2.5 \\\\ \\text{Similarly, }\\cos (30{}^\\circ )=\\dfrac{x}{5}\\\\ x=5\\left( \\dfrac{\\sqrt{3}}{2} \\right) \\\\ \\therefore x=4.33 $$ ","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Vector components Quiz 1","text":" If vector $$\\overrightarrow{{{F}_{1}}}=5\\hat{i}+6\\hat{j}+2\\hat{k}$$ represents the force exerted by object 1 on a table and $$\\overrightarrow{{{F}_{2}}}=7\\hat{j}+3\\hat{k}$$ is the force exerted by the 2nd object on the table. Calculate the magnitude of the combined force exerted by both the objects on the table. ","comment":{"@type":"Comment","text":"The magnitude of the vector is given as the square root of the sum of the squares of its vector components. "},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 12.5","position":1},{"@type":"Answer","encodingFormat":"text/html","text":"10.5","position":2},{"@type":"Answer","encodingFormat":"text/html","text":" 4.6","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 14.7","position":0,"answerExplanation":{"@type":"Comment","text":" $$\\text{The combined force will be,}\\\\ \\overrightarrow{F}=\\overrightarrow{{{F}_{1}}}+\\overrightarrow{{{F}_{2}}}\\\\ \\Rightarrow\\overrightarrow{F}=5\\hat{i}+13\\hat{j}+5\\hat{k} \\\\ \\Rightarrow |\\overrightarrow{F}|=\\sqrt{25+169+25} \\\\ \\Rightarrow |\\overrightarrow{F}|=\\sqrt{219} \\\\ \\therefore |\\overrightarrow{F}|=14.7 $$ ","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Vector components Quiz 1","text":"If the $$x, y$$ and $$z$$ projection of a vector is 3 along x-axis, 4 along y-axis and 0 along z-axis. Calculate the length of the vector from the origin. ","comment":{"@type":"Comment","text":"The length of the vector is obtained by calculating the magnitude of the vector. "},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 4","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" 3","position":2},{"@type":"Answer","encodingFormat":"text/html","text":" 6","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 5","position":1,"answerExplanation":{"@type":"Comment","text":" The vector framed from the given projections is, $$3\\hat{i}+4\\hat{j}+0\\hat{k} $$ $$ \\\\ $$ The length of the vector is obtained by calculating the magnitude of the vector. $$ \\\\ $$ So, the length of the vector is, $$ \\\\ \\therefore\\sqrt{{{3}^{2}}+{{4}^{2}}+{{0}^{2}}}=5$$ ","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]}]}