In one sentence, describe what the following article is about: Write down all the information you have concerning the two vectors. We'll assume you only have the vector's definition in terms of its dimensional coordinates (also called components). If you already know a vector's length (its magnitude), you'll be able to skip some of the steps below. Example: The two-dimensional vector u{\displaystyle {\overrightarrow {u}}} = (2,2). Vector v{\displaystyle {\overrightarrow {v}}} = (0,3). These can also be written as u{\displaystyle {\overrightarrow {u}}} = 2i + 2j and v{\displaystyle {\overrightarrow {v}}} = 0i + 3j = 3j. While our example uses two-dimensional vectors, the instructions below cover vectors with any number of components. To find the angle between two vectors, start with the formula for finding that angle's cosine. You can learn about this formula below, or just write it down: cos = (u{\displaystyle {\overrightarrow {u}}} v{\displaystyle {\overrightarrow {v}}}) / (||u{\displaystyle {\overrightarrow {u}}}|| ||v{\displaystyle {\overrightarrow {v}}}||) ||u{\displaystyle {\overrightarrow {u}}}|| means "the length of vector u{\displaystyle {\overrightarrow {u}}}." u{\displaystyle {\overrightarrow {u}}} v{\displaystyle {\overrightarrow {v}}} is the dot product (scalar product) of the two vectors, explained below. Picture a right triangle drawn from the vector's x-component, its y-component, and the vector itself. The vector forms the hypotenuse of the triangle, so to find its length we use the Pythagorean theorem. As it turns out, this formula is easily extended to vectors with any number of components. ||u||2 = u12 + u22. If a vector has more than two components, simply continue adding +u32 + u42 + ... Therefore, for a two-dimensional vector, ||u|| = (u12 + u22). In our example, ||u{\displaystyle {\overrightarrow {u}}}|| = (22 + 22) = (8) = 22. ||v{\displaystyle {\overrightarrow {v}}}|| = (02 + 32) = (9) = 3. You have probably already learned this method of multiplying vectors, also called the scalar product.To calculate the dot product in terms of the vectors' components, multiply the components in each direction together, then add all the results. For computer graphics programs, see Tips before you continue. Remember, cos = (u{\displaystyle {\overrightarrow {u}}} v{\displaystyle {\overrightarrow {v}}}) / (||u{\displaystyle {\overrightarrow {u}}}|| ||v{\displaystyle {\overrightarrow {v}}} ||). Now you know both the dot product and the lengths of each vector. Enter these into this formula to calculate the cosine of the angle. You can use the arccos or cos-1 function on your calculator to find the angle from a known cos value. For some results, you may be able to work out the angle based on the unit circle.
Summary: Identify the vectors. Write down the cosine formula. Calculate the length of each vector. Calculate the dot product of the two vectors. Plug your results into the formula. Find the angle based on the cosine.

If you're not already signed in to your Reddit account, enter your login information to do so now. It's near the top-right corner of Reddit. This opens your current profile. At the top of the main column is a blue bar that advertises Reddit's new profile experience. It's in the blue bar at the top of the screen. This agreement informs you of some new features. Clicking the box indicates that you've read the agreement and understand that converting your profile to the new format is permanent. It's at the bottom of the agreement. This opens the settings page for your new profile. It's the blue button under the "Profile Image" header. This launches your computer's file browser. If you don't see it in the current folder, navigate to the folder where it's saved. This will upload the photo to your profile. You can also upload a cover photo for the top of your profile. To do this, click Upload Photo under the "Profile Cover" header, then select the photo you want to upload. Now that you've converted your profile, you can enter a display name and some information about yourself into the "About" box. This is optional. It's near the top-left corner of the current page. Your profile image (and cover photo, if you chose one) is now saved to your profile.
++++++++++
One-sentence summary -- Go to https://www.reddit.com in a web browser. Click your username. Click Learn More. Read the agreement and click the check box. Click Give Me the New Profile. Click Upload Photo. Double-click the photo you want to upload. Click Back to your Profile.

Problem: Article: Line an 8-inch or 9-inch (20.32 or 22.86 centimeters) square baking pan with wax paper or parchment paper going each direction so that the bottom and all four sides are covered. Leave about 2 inches (5.08 centimeters) of wax paper hanging over the edges. This way, you can use them like handles and pull the finished fudge out. You can also do this using tin foil instead. Stir the ingredients often to help the chips melt evenly. Keep stirring until everything is melted, smooth, and mixed together. There should be no lumps, clumps, streaks, or swirls. If you do not like peanut butter, you can use another type of chip, such as mint chocolate chip, red velvet chocolate chip, etc. The vanilla will help add a more complex flavor to your fudge. The salt will help cut the sweetness as well as bring out the chocolate and peanut butter flavors. Use a spatula to help guide the mixture across the bottom of the pan. Once everything is in the pan, place it into the fridge and wait until it becomes firm. The fudge must be firm before you cut it, or you'll end up with an oozy mess. Once the fudge has hardened up, use the wax paper handles to lift it out of the pan. Peel the wax paper away from the edges. If the fudge is too hard to cut, let it sit on the kitchen counter for 5 minutes. This will allow it to soften a little. Make sure that you are using a sharp knife to do this. If there are any leftovers, cover them up and store them in the fridge. The fudge should last about 1 week.
Summary:
Prepare your baking pan. Heat the condensed milk, chocolate chips, and peanut butter chips in a saucepan over low heat until melted. Remove the saucepan from heat and stir in the vanilla extract and salt. Pour the mixture into the lined pan and refrigerate for at least 2 hours. Remove the fudge from the pan. Cut the fudge into 1-inch (2.54 centimeters) pieces. Serve the fudge on a plate.