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Two vectors are equivagraphict if they have the same magnitude and the same direction. Let u u 1 u 2 and v v 1 v 2 . Then u 1 u 2 v 1 v 2 if and only if u 1 v 1 and u 2 v 2. Operations on Vectors. To multiply a vector v by a positive real number we multiply its graphicgth by the number. Its direction stays the same. When a vector v is multiplied by 2 for instance its graphicgth is 3. Cross product De nition 3.1. Let vand wbe two vectors in R3. The cross product of vand w denoted v w is the vector de ned as follows the graphicgth of v wis the area of the parallelogram with sides v and w that is kvkkwksin . v wis orthogonal to both vand w. the three vectors v w and v w form a right-handed set of Thanks to all of you who support me on Patreon. You da real mvps 1 per month helps ) httpswww.patreon.compatrickjmt Finding a Unit Vector Ex 1. In this

Adding vectors in magnitude and direction form. Vector components from magnitude & direction (advanced) Review your knowledge of vector magnitude and direction and use them to solve problems. If youre seeing this message it means were having trouble loading external resources on our website. If youre behind a web filter please make sure that the domains .kastatic.org and .kasandbox.org To distinguish between scalars and vectors we will denote scalars by lower case italic type such as a b c etc. and denote vectors by lower case boldface type such as u v w etc. In handwritten script this way of distinguishing between vectors and scalars must be modified. It is customary to leave scalars as regular hand written script and 9.4 20 Let v 5j and let u be a vector with graphicgth 3 that starts at the origin and rotates in the xy-plane. Find the maximum and minimum values of the graphicgth of the vector u v. In what direction does u v point [Solution] Rotating a vector with graphicgth 3 that starts at the origin in the xy-plane forms a circle with center (00) and radius 3

Since this is a quadrant 3 vector the direction is . v w (8 5) has direction given by . Since this is a quadrant 1 vector is the direction of v w. In the next section we will diagram both the addition and subtraction of these two vectors. Notice that v w is the diagonal of a parallelogram formed by v and w. graphicignment 3. Solutions. Problems. February 22. 1. Find a vector of magnitude 3 in the direction opposite to the direction of v 1 2 i 1 2 j 1 2 k. Solution. Finding Vector Components Let u and v be nonzero vectors such that u w 1 w 2 where w 1 and w 2 are orthogonal and w 1 is parallel to (or a scalar multiple of) v. The vectors w 1 and w 2 are called vector components of u. The vector w 1 is the projection of u onto v and is denoted by w 1 proj v u. The vector w 2 is given by w 2 u - w 1 Let u -4 -3. Find the unit vector in the direction of u and write your answer in component form.