VIDEO solution: Determine the equation of the line of intersection of planes 1 and 2.
Express your answer in the form: [x,y,z] = [xo,yo,0] + t[a, b,1].
Find the point where the line crosses the xz-plane.
[5 marks, 2 marks]
T1: 2x + 2y - 3z + 7 = 0
T2: 4 (2024)
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Make an ajax call to the server and get the search database. let databaseUrl = `/search/whiletype_database/`; let resp = single_whiletyping_ajax_promise; if (resp === null) { whiletyping_database_initial_burst = whiletyping_database_initial_burst + 1; single_whiletyping_ajax_promise = resp = new Promise((resolve, reject) => { $.ajax({ url: databaseUrl, type: 'POST', data:{csrfmiddlewaretoken: "vD5mhBkZIDkdOUIPCIBgrYE7njQ8HMEQREvWhXskQTEZpBPzBdkouXQOZ6cd8whv"}, success: function (data) { // 3. verify that the elements of the database exist and are arrays if ( ('books' in data) && ('curriculum' in data) && ('topics' in data) && Array.isArray(data.books) && Array.isArray(data.curriculum) && Array.isArray(data.topics)) { localforage.setItem('whiletyping_last_success', (new Date()).getTime()); localforage.setItem('whiletyping_database', data); resolve(data); } }, error: function (error) { console.log(error); resolve(null); }, complete: function (data) { single_whiletyping_ajax_promise = null; } }) }); } return resp; } return Promise.resolve(null); }).catch(function(err) { console.log(err); return Promise.resolve(null); }); } function get_whiletyping_search_object() { // gets the fuse objects that will be in charge of the search if (whiletyping_search_object){ return Promise.resolve(whiletyping_search_object); } database_promise = localforage.getItem('whiletyping_database').then(function(database) { return localforage.getItem('whiletyping_last_success').then(function(last_success) { if (database==null || (new Date()) - (new Date(last_success)) > 1000*60*60*24*30 || (new Date('2023-04-25T00:00:00')) - (new Date(last_success)) > 0) { // New database update return get_whiletyping_database().then(function(new_database) { if (new_database) { database = new_database; } return database; }); } else { return Promise.resolve(database); } }); }); return database_promise.then(function(database) { if (database) { const options = { isCaseSensitive: false, includeScore: true, shouldSort: true, // includeMatches: false, // findAllMatches: false, // minMatchCharLength: 1, // location: 0, threshold: 0.2, // distance: 100, // useExtendedSearch: false, ignoreLocation: true, // ignoreFieldNorm: false, // fieldNormWeight: 1, keys: [ "title" ] }; let curriculum_index={}; let topics_index={}; database.curriculum.forEach(c => curriculum_index[c.id]=c); database.topics.forEach(t => topics_index[t.id]=t); for (j=0; j
Solutions
Textbooks
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Take the cross product of the normal vectors, ⃑ 𝑑 = ⃑ 𝑛 × ⃑ 𝑛 , to give a vector, ⃑ 𝑑 , parallel to the line of intersection between the planes. The vector equation of the line of intersection is then given by ⃑ 𝑟 = ⃑ 𝑟 + 𝑡 ⃑ 𝑑 , where 𝑡 is a scalar.
The parametric equations of the line are x = 1 − 2t, y = 3t and z = −1. The point of intersection will satisfy the equation of the plane for some value of the parameter t. Substitute the parametric equations into the equation of the plane and solve for t.When t = −1, the line intersects the plane.
Two straight lines will intersect at a point if they are not parallel. The point of intersection is the meeting point of two straight lines. If two crossing straight lines have the same equations, the intersection point can be found by solving both equations at the same time.
Solution: In three dimensions (which we are implicitly working with here), what is the intersection of two planes? As long as the planes are not parallel, they should intersect in a line. So our result should be a line.
If two planes intersect, then their intersection is a line (Postulate 6). A line contains at least two points (Postulate 1). If two lines intersect, then exactly one plane contains both lines (Theorem 3). If a point lies outside a line, then exactly one plane contains both the line and the point (Theorem 2).
Intercepts of the plane are points on the x, y and z axis where the plane cuts their axes. If the x-intercept of the plane is (a, 0, 0), y-intercept is (0, b, 0) and z-intercept is (0, 0, c) then the equation of the plane is x/a + y/b + z/c = 1. Equation (1) represents the equation of the plane in intercept form.
Given that a perpendicular line to a plane can only intersect the plane at one point, the intersection of a plane and a line perpendicular to it is a single point.
We can observe that there can be many straight lines passing through the point 'A'. And there are many lines passing through the point 'B'. But there is one and only one straight line which is passing through the point 'A' as well as point 'B'. Therefore, two distinct points always determine a unique straight line.
In a geometric context, two distinct points P_1and P_2 always determine a unique line in the Cartesian plane (this is one of the basic postulates of Euclidean geometry). Only the non-vertical lines, however, can be described by the graph of a function f(x) = mx + b.
The direction vector of the line of intersection of two planes is parallel to the cross product of the normal vectors to the planes. That is because, the line of intersection is in both planes, so it is perpendicular on the two normal vectors to the planes, which is the cross product vector.
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