cartesian and vector equation of line perpendicular to given lines

Описание: cartesian and vector equation of line perpendicular to given lines
Find the equation of a line in vector and cartesian form which passes through the point (1,-2,4) and is perpendicular to the lines
[(x-8)/3]=[(y+19)/(-16)]=[(z-10)/7]
and
r =15i+29j+5k +t(3i+8j-5k)

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In this video the problem of finding the equation of a line in both vector form and cartesian form is explained step by step using a worked example. The task is to determine the equation of a line passing through the point (1, −2, 4) that is perpendicular to the line given in symmetric form as (x − 8)/3 = (y + 19)/(−16) = (z − 10)/7. This is one of the most important applications of vectors and three dimensional geometry, forming a foundation for exams across school boards, advanced levels, and university engineering mathematics.

The method begins with identifying the direction ratios of the given line. These are read directly from the denominators in the symmetric form. The concept of perpendicularity in three dimensions is then applied a standard tool for orthogonality in vector spaces. With the condition satisfied, the required line can be written through the given point in vector form. The equivalent cartesian form is then obtained, which is a required skill for many examination settings where answers must be presented in both representations.

This type of problem is a regular feature of CBSE Class 12 Mathematics, specifically under the chapters Vector Algebra and Three Dimensional Geometry. Past CBSE board papers repeatedly test vector and cartesian equations of lines and perpendicularity conditions. It also appears in ICSE Class 12 Mathematics, where similar examples are asked in Section B and Section C long answer questions. Students across Indian state boards also meet this problem type under the same chapters.

In international curricula, this topic is fully tested in Cambridge International A Level Mathematics (9709) where Paper 3 Pure Mathematics includes vectors in three dimensions, and in Edexcel and AQA A Level Mathematics under vectors in 3D. Past papers from these boards frequently include questions on writing vector and cartesian forms of lines and checking perpendicularity. In the IB Mathematics HL and SL syllabi, the topic of vectors in three dimensions requires students to work with equations of lines, direction vectors, and perpendicularity. Previous IB papers show this exact type of line construction as a core exam question.

For competitive examinations, IIT JEE Main and Advanced consistently ask problems in vectors and 3D geometry where a line must be constructed perpendicular to a given line or lying in relation to a plane. Equations in both vector and cartesian form are required. These problems have been asked in multiple past year papers with almost identical structure to the one solved in this video. Mastery of the method ensures efficiency in answering such high-stakes exam questions.

At the higher education level, this problem is taught in BTech and BE Engineering Mathematics courses in the first year across universities worldwide. Engineering Mathematics 1 and 2 cover vector algebra, direction cosines, vector equations of lines, and applications of perpendicularity in structural mechanics, robotics, computer graphics, and civil and mechanical engineering problems. University exam papers frequently include tasks requiring students to write vector and cartesian equations of lines with given conditions.

This worked solution will guide you through the entire process, reinforcing understanding of direction vectors and the ability to move between vector and cartesian forms. It is relevant for board examinations like CBSE and ICSE, international qualifications like Cambridge A Levels and IB Mathematics, competitive tests like IIT JEE, and university-level engineering mathematics in BTech and BE courses. Practicing these problems regularly ensures fluency, accuracy, and confidence in solving exam style questions that appear in past papers across these curricula.

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