- \r\n \t
- Perform vector addition and scalar multiplication.<\/li>\r\n \t
- Perform operations with vectors in terms of i and j .<\/li>\r\n \t
- Find the dot product of two vectors.<\/li>\r\n<\/ul>\r\n<\/section>\r\n
Combining Vectors<\/h2>\r\n
\r\n\r\nThe Main Idea\u00a0<\/strong>\r\nVectors show up constantly in real life\u2014think about a boat crossing a river (engine pushing one way, current pushing another) or a quarterback throwing a football (force and angle combined). To solve these problems, you need to know how to combine vectors.<\/p>\r\n
Scalar Multiplication: Stretching and Flipping<\/strong><\/p>\r\n
When you multiply a vector [latex]\\mathbf{v}[\/latex] by a scalar [latex]k[\/latex], you change its length and possibly its direction:<\/p>\r\n\r\n
- \r\n \t
- If [latex]k > 0[\/latex]<\/strong>: The vector stretches or shrinks but keeps the same direction. For example, [latex]2\\mathbf{v}[\/latex] doubles the length; [latex]0.5\\mathbf{v}[\/latex] halves it.<\/li>\r\n \t
- If [latex]k < 0[\/latex]<\/strong>: The vector flips to point the opposite direction. For instance, [latex]-\\mathbf{v}[\/latex] has the same length as [latex]\\mathbf{v}[\/latex] but points the other way.<\/li>\r\n \t
- If [latex]k = 0[\/latex]<\/strong>: You get the zero vector [latex]\\mathbf{0}[\/latex].<\/li>\r\n<\/ul>\r\n
Vector Addition: Two Methods, Same Result<\/strong><\/p>\r\n
To add vectors [latex]\\mathbf{v} + \\mathbf{w}[\/latex], use either approach:<\/p>\r\n\r\n
- \r\n \t
- Triangle Method<\/strong>: Place the starting point of [latex]\\mathbf{w}[\/latex] at the endpoint of [latex]\\mathbf{v}[\/latex]. The sum runs from [latex]\\mathbf{v}[\/latex]'s start to [latex]\\mathbf{w}[\/latex]'s end. Think of it like walking from A to B, then B to C\u2014your total displacement goes directly from A to C.<\/li>\r\n \t
- Parallelogram Method<\/strong>: Start both vectors at the same point and complete the parallelogram. The diagonal is your sum.<\/li>\r\n<\/ul>\r\n
Order doesn't matter: [latex]\\mathbf{v} + \\mathbf{w} = \\mathbf{w} + \\mathbf{v}[\/latex] (commutative property).<\/p>\r\n
Triangle Inequality<\/strong>: The sum's length satisfies [latex]||\\mathbf{v} + \\mathbf{w}|| \\leq ||\\mathbf{v}|| + ||\\mathbf{w}||[\/latex]. Equality only happens when vectors point the same direction.<\/p>\r\n
Vector Subtraction: Adding the Opposite<\/strong><\/p>\r\n
For [latex]\\mathbf{v} - \\mathbf{w}[\/latex], rewrite it as [latex]\\mathbf{v} + (-\\mathbf{w})[\/latex]. Graphically, the difference points from [latex]\\mathbf{w}[\/latex]'s endpoint to [latex]\\mathbf{v}[\/latex]'s endpoint.<\/p>\r\n\r\n<\/div>\r\n
Using vectors [latex]\\bf{w}[\/latex] and [latex]\\bf{w}[\/latex] from Example: Combining Vectors, sketch the vector [latex]\\bf{2w - v}[\/latex].\r\n [reveal-answer q=\"fs-id1167793933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793933124\"][caption id=\"attachment_3477\" align=\"aligncenter\" width=\"270\"]![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/28191840\/2-1-tryitans2.jpeg\")
The graph of the combined vector [latex]\\bf{2w - v}[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<\/section>Watch the following video to see the worked solution to the above example.\r\n - Parallelogram Method<\/strong>: Start both vectors at the same point and complete the parallelogram. The diagonal is your sum.<\/li>\r\n<\/ul>\r\n
- If [latex]k < 0[\/latex]<\/strong>: The vector flips to point the opposite direction. For instance, [latex]-\\mathbf{v}[\/latex] has the same length as [latex]\\mathbf{v}[\/latex] but points the other way.<\/li>\r\n \t
- If [latex]k > 0[\/latex]<\/strong>: The vector stretches or shrinks but keeps the same direction. For example, [latex]2\\mathbf{v}[\/latex] doubles the length; [latex]0.5\\mathbf{v}[\/latex] halves it.<\/li>\r\n \t