{"id":20037,"date":"2022-09-21T08:06:36","date_gmt":"2022-09-21T08:06:36","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=7838"},"modified":"2022-09-21T08:06:36","modified_gmt":"2022-09-21T08:06:36","slug":"examples-4","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/09\/21\/examples-4\/","title":{"rendered":"Examples"},"content":{"rendered":"\n

Example 4<\/strong>: Find a unit vector v<\/strong> with the same direction as the vector u<\/strong> given that u<\/strong> = \u27e87, \u2212 1\u27e9.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Two special unit vectors, i<\/strong> = \u27e81, 0\u27e9 and j<\/strong> = \u27e80, 1\u27e9, can be used to express any vector v<\/strong> = \u27e8a, b\u27e9<\/em>.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example 5:<\/strong> Write u<\/strong> = \u27e85, 3\u27e9 in terms of the i<\/strong> and j<\/strong> unit vectors (Figure 5 ).<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Figure 5 <\/strong>
             Drawing for Example 5.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Vectors exhibit algebraic properties similar to those of real numbers (Table  1).<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example 6:<\/strong> Find 4 u<\/strong> + 5 v<\/strong> if u<\/strong> = 7 i<\/strong> \u2212 3 j<\/strong> and v<\/strong> = \u22122 i<\/strong> + 5 j<\/strong>.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Given two vectors, u<\/strong> = \u27e8 a, b\u27e9<\/em> = a<\/em> i<\/strong> + b<\/em> j<\/strong> and v<\/strong> = \u27e8c, d\u27e9<\/em> = c<\/em> i<\/strong> + d<\/em> j<\/strong>, the dot product<\/strong>, written as u<\/strong>\u00b7 v<\/strong>, is the scalar quantity u<\/strong> \u02d9 v<\/strong> = ac + bd<\/em>. If u, v<\/strong>, and w<\/strong> are vectors and q<\/em> is a real number, then dot products exhibit the following properties:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

The last property, u \u02d9 v<\/strong> = | u<\/strong>| | v<\/strong>| cos \u03b1, can be used to find the angle between the two nonzero vectors u<\/strong> and v<\/strong>. If two vectors are perpendicular to each other and form a 90\u00b0 angle, they are said to be orthogonal<\/strong>. Because cos 90\u00b0 = 0, the dot product of any two orthogonal vectors is 0.<\/p>\n\n\n\n

Example 7:<\/strong> Given that u<\/strong> = \u27e8 5<\/em>, \u22123\u27e9 and v<\/strong> = \u27e86, 10\u27e9, show that u<\/strong> and v<\/strong> are orthogonal by demonstrating that the dot product of u<\/strong> and v<\/strong> is equal to zero.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example 8:<\/strong> What is the angle between u = \u27e85, \u22122\u27e9 and v = \u27e86, 11\u27e9?<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

An object is said to be in a state of static equilibrium<\/strong> if all the force vectors acting on the object add up to zero.<\/p>\n\n\n\n

Example 9:<\/strong> A tightrope walker weighing 150 pounds is standing closer to one end of the rope than the other. The shorter length of rope deflects 5\u00b0 from the horizontal. The longer length of rope deflects 3\u00b0. What is the tension on each part of the rope?<\/p>\n\n\n\n

Draw a force diagram with all three force vectors in standard position (Figure  6).<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Figure 6 <\/strong>
              Drawing for Example 9.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

The sum of the force vectors must be zero for each component.<\/p>\n\n\n\n

For the i<\/strong> component: \u2212 | u<\/strong>|cos 5\u00b0 + | v<\/strong>| cos 3\u00b0 = 0<\/p>\n\n\n\n

For the j<\/strong> component: | u<\/strong>| sin5\u00b0 + |v| cos 3\u00b0 \u2212 150 =<\/p>\n\n\n\n

Solve these two equations for | u<\/strong>| and | v<\/strong>|:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Substituting the values for the sines and cosines:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Multiply the first equation by 0.0872 and the second by 0.9962:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Add the two equations and solve for | v<\/strong>|:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Substitute and solve for | u<\/strong>|:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Example 4: Find a unit vector v with the same direction as the vector u given that u = \u27e87, \u2212 1\u27e9. Two special unit vectors, i = \u27e81, 0\u27e9 and j = \u27e80, 1\u27e9, can be used to express any vector v = \u27e8a, b\u27e9. Example 5: Write u = \u27e85, 3\u27e9 in terms of the i and j unit vectors (Figure 5 ). Figure 5              Drawing for Example 5. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[716],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/20037"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=20037"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/20037\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=20037"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=20037"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=20037"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}