{"id":20257,"date":"2024-04-15T05:50:29","date_gmt":"2024-04-15T05:50:29","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20257"},"modified":"2024-04-15T05:50:29","modified_gmt":"2024-04-15T05:50:29","slug":"for-a-position-vector-rmr-rmxhat-i-rmyhat-j-rmzhat-k-the-norm-of-the-vector-can-be-defined-as-left-overrightarrow-textr-right-sqrt","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/for-a-position-vector-rmr-rmxhat-i-rmyhat-j-rmzhat-k-the-norm-of-the-vector-can-be-defined-as-left-overrightarrow-textr-right-sqrt\/","title":{"rendered":"For a position vector \\[{\\rm{r}} = {\\rm{x\\hat i}} + {\\rm{y\\hat j}} + {\\rm{z\\hat k}}\\] the norm of the vector can be defined as $$\\left| {\\overrightarrow {\\text{r}} } \\right| = \\sqrt {{{\\text{x}}^2} + {{\\text{y}}^2} + {{\\text{z}}^2}} .$$ Given a function $$\\phi = \\ln \\left| {\\overrightarrow {\\text{r}} } \\right|,$$ its gradient $$\\nabla \\phi $$ is A. $$\\overrightarrow {\\text{r}} $$ B. $$\\frac{{\\overrightarrow {\\text{r}} }}{{\\left| {\\overrightarrow {\\text{r}} } \\right|}}$$ C. $$\\frac{{\\overrightarrow {\\text{r}} }}{{\\overrightarrow {\\text{r}} \\cdot \\overrightarrow {\\text{r}} }}$$ D. $$\\frac{{\\overrightarrow {\\text{r}} }}{{{{\\left| {\\overrightarrow {\\text{r}} } \\right|}^3}}}$$"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;$$\\overrightarrow {\\text{r}} $$&#8221; option2=&#8221;$$\\frac{{\\overrightarrow {\\text{r}} }}{{\\left| {\\overrightarrow {\\text{r}} } \\right|}}$$&#8221; option3=&#8221;$$\\frac{{\\overrightarrow {\\text{r}} }}{{\\overrightarrow {\\text{r}} \\cdot \\overrightarrow {\\text{r}} }}$$&#8221; option4=&#8221;$$\\frac{{\\overrightarrow {\\text{r}} }}{{{{\\left| {\\overrightarrow {\\text{r}} } \\right|}^3}}}$$&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{{\\overrightarrow {\\text{r}} }}{{\\left| {\\overrightarrow {\\text{r}} } \\right|}}}$.<\/p>\n<p>The gradient of a scalar field $\\phi$ is a vector field that points in the direction of the greatest rate of increase of $\\phi$, and its magnitude is equal to the magnitude of the rate of increase of $\\phi$. In other words, the gradient of $\\phi$ is the vector field that tells you how fast $\\phi$ is changing at each point in space.<\/p>\n<p>In this case, the scalar field is $\\phi = \\ln \\left| {\\overrightarrow {\\text{r}} } \\right|$, which is the natural logarithm of the magnitude of the position vector. The rate of change of $\\phi$ in the direction of $\\overrightarrow {\\text{r}}$ is $\\frac{\\partial \\phi}{\\partial \\overrightarrow {\\text{r}}} = \\frac{\\overrightarrow {\\text{r}} }{{\\left| {\\overrightarrow {\\text{r}} } \\right|}}$. Therefore, the gradient of $\\phi$ is $\\nabla \\phi = \\frac{{\\overrightarrow {\\text{r}} }}{{\\left| {\\overrightarrow {\\text{r}} } \\right|}}$.<\/p>\n<p>The other options are incorrect because they do not point in the direction of the greatest rate of increase of $\\phi$. For example, $\\overrightarrow {\\text{r}}$ does not point in the direction of the greatest rate of increase of $\\phi$ because it is not always in the same direction as the gradient of $\\phi$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;$$\\overrightarrow {\\text{r}} $$&#8221; option2=&#8221;$$\\frac{{\\overrightarrow {\\text{r}} }}{{\\left| {\\overrightarrow {\\text{r}} } \\right|}}$$&#8221; option3=&#8221;$$\\frac{{\\overrightarrow {\\text{r}} }}{{\\overrightarrow {\\text{r}} \\cdot \\overrightarrow {\\text{r}} }}$$&#8221; option4=&#8221;$$\\frac{{\\overrightarrow {\\text{r}} }}{{{{\\left| {\\overrightarrow {\\text{r}} } \\right|}^3}}}$$&#8221; correct=&#8221;option1&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20257","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>For a position vector \\[{\\rm{r}} = {\\rm{x\\hat i}} + {\\rm{y\\hat j}} + {\\rm{z\\hat k}}\\] the norm of the vector can be defined as $$\\left| {\\overrightarrow {\\text{r}} } \\right| = \\sqrt {{{\\text{x}}^2} + {{\\text{y}}^2} + {{\\text{z}}^2}} .$$ Given a function $$\\phi = \\ln \\left| {\\overrightarrow {\\text{r}} } \\right|,$$ its gradient $$\\nabla \\phi $$ is A. $$\\overrightarrow {\\text{r}} $$ B. $$\\frac{{\\overrightarrow {\\text{r}} }}{{\\left| {\\overrightarrow {\\text{r}} } \\right|}}$$ C. $$\\frac{{\\overrightarrow {\\text{r}} }}{{\\overrightarrow {\\text{r}} \\cdot \\overrightarrow {\\text{r}} }}$$ D. $$\\frac{{\\overrightarrow {\\text{r}} }}{{{{\\left| {\\overrightarrow {\\text{r}} } \\right|}^3}}}$$<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/for-a-position-vector-rmr-rmxhat-i-rmyhat-j-rmzhat-k-the-norm-of-the-vector-can-be-defined-as-left-overrightarrow-textr-right-sqrt\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"For a position vector \\[{\\rm{r}} = {\\rm{x\\hat i}} + {\\rm{y\\hat j}} + {\\rm{z\\hat k}}\\] the norm of the vector can be defined as $$\\left| {\\overrightarrow {\\text{r}} } \\right| = \\sqrt {{{\\text{x}}^2} + {{\\text{y}}^2} + {{\\text{z}}^2}} .$$ Given a 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